This is a frame challenge to the question.
Why does the question assume that there can not be worlds habitable for liquid water using life around a red dwarf star? There are some factors which may make liquid water using life improbable around many or most red dwarf stars, but that is not certain at the moment.
Furthermore, all other types of stars should also be suitable for having planets which are the right temperatures for hypothetical ammonia using life.
The question seems to assume that all planets orbit their stars at the same distances. Thus if a star is very luminous all of its planets will be very, very hot, too hot for liquid water based life, and if a star is very dim like a red dwarf star all of its planets will all be very, very cold, too cold for liquid water based life.
Take a look at our own solar system, which has eight officially recognized planets at the moment. The semi-major axis of the orbit of the Earth around the Sun is defined as one Astronomical Unit, or AU.
So according to this table, https://www.encyclopedia.com/reference/encyclopedias-almanacs-transcripts-and-maps/major-planets-solar-system-table and not allowing for the eccentricity of planetary orbits:
Mercury orbits the Sun (the source of light and heat), at 0.39 AU, Venus at 0.72 AU, Earth at 1 AU, Mars at 1.52 AU, Jupiter at 5.20 AU, Saturn at 9.54 AU, Uranus at 19.2 AU, and Neptune at 30.06 AU.
So Neptune is 77.076 times as far from the Sun as Mercury is. Therefore Mercury receives 77.076 squared, or 5.940.85, times as much radiation from the Sun as Neptune does.
Radiation from the Sun is not the only source of planetary heat, but it is the major source, and differences in the amount of radiation received from the Sun is the major cause of differences in planetary temperatures.
At the present time in 2021 over 4,000 exoplanets in other star systems have been discovered, and astronomers have found that they vary a lot in various aspects.
Wikipedia has a list of exoplanet extremes.
Planet COCONUTS-2b orbits around the star COCONUTS-2 at so great a distance (about 6,471 AU) that its year is estimated to be about 1,100,000 Earth years long.
The previous record holder for longest year, JMASS J2126-8140, is a giant planet or brown dwarf orbiting the star TYC 9481-927-1, a spectral type M1V, a red dwarf, at a distance of about 6,900 AU and with a year about 900,000 Earth years long.
The record for the coldest exoplanet is held by OGLE-2016-BLG-1195Lb, which orbits the very dim star OGLE-2016-BLG-1195L, at about 1 AU distance, but has a temperature of only about 31 degreees K.
There is a class of "Hot Jupiters", very large planets which orbit very close to their stars and are very hot as a result. "Ultra-hot Jupiters" have daytime temperatures over 2,200 degrees K.
Hot Jupiters have very close obits but are not common around the least massive and least luminous stars, red dwarfs.
They appear to be more common around F- and G-type stars and less so around K-type stars. Hot Jupiters around red dwarfs are very rare.5 Generalizations about the distribution of these planets must take into account the various observational biases, but in general their prevalence decreases exponentially as a function of the absolute stellar magnitude.6
Many hot jupiters orbit stars that are brighter than the Sun at distances much closer than the orbit of Mercury in our solar system.
So in our solar system the orbital distances and surface temperatures of planets vary greatly, and examples of planets in other star systems show that the orbits of planets do not have to be proportional to the mass and luminosity of their stars. Any type of star, from the brightest to the dimmest, can have close planets, even planets which are many times closer than Mercury is to the Sun. Any type of star, from the brightest to the dimmest, can have distant planets, even planets which are many times as distant as Neptune is from the Sun.
So any specific star of any type could possibly have one or more planets orbiting within its circumstellar habitable zone. And no specific star of any type is guaranteed to have even a single planet within its circumstellar habitable zone, at least not until one or more planets are actually discovered with that zone.
Wikipedia has an article discussing Hypothetical biochemestries which myhypothetical be posisble in enviroments with different chemicals present in abundance, or with different temperatures. That is a good place to star reseraching ideas oa bout different biochemestries for planets where liquid water based life could not exist.
The circumstellar habitable zone of a star is the range of distances where the temperatures of hypothetical planets would enable water to be liquid on their surfaces. It is shaped like a hollow shell between two concentric spheres with different radii. But since most star systems would have their planets orbiting in almost exactly the same plane, with no more than a few degrees of tilt, it is common to picture a circumstellar habitable zone as a disc with a smaller disc cut out from the center.
And if alternate forms of biochemestry can work and be used by lifeforms at temperatures whee water is a gas or a solid, a star would have several different concentric habitable zones for different biochestries requring different temperatures.
Even if there are several possible habitable zones around stars, no doubt many stars have planets which orbit outside of any habitable zone.
Those planets might orbit too close and too hot for any possible form of life. They might orbit too far and too cold for any possible form of life. Or they might orbit within gaps betwen two habitable zones, where no possible lifeforms could survive the intermediate temperatures.
And possibly some stars could have planets orbiting within every possible habitable zone.
In theory it is easy to determine the inner and outer limits of the circumstellar habitable zone for liquid water using life forms of any specific stars whose luminiosity relative to that of the Sun is known. Just adjust the inner and outer edges of the Sun's circumstellar habitable zone to account for the star's luminosity.
But the inner and outer edges of the Sun's circumstellar habitable zone are not know with certainty.
The table lists of number of widely varying estimates of the inner and outer edges, or both, of the Sun's circumstellar habitable zone.
I note that human beings cannot survive unprotected by clothing, shelters, and sometimes breathng equipment, in large parts of the biosphere of Earth where other lifeforms can survive and flourish. Also Earth formed about 4.54 billion years ago, and the earliest certainly known life on Earth was 3 billion years ago, but Earth didn't have an atmosphere breathable for humans until about 540 million years ago.
So most scientific discussions of planetary habitability mean habitability for liquid water using lifeforms in general, and only a minority of such habitable worlds would also have oxygen rich atmospheres for land animals to breathe.
Some of the inner and outer extensions to the habitable zone require specific atmospheric compositions to maintain the proper temperatures. Those atmospheres might not be breathable for humans, but native lifeforms might have no trouble tolerating them.
I added a lot more to this answer, but lost all of that work on it.
Tues. 09-28-2021 I guess I'll try to complete the answer.
Of course the definition of the habitable zone is a zone where planets (with dense enough atmospheres that water doesn't go diectly from ice to vapor) can have the right temperatures for liquid water on their surfaces.
There are many objects in the outer solar system which are largely composed of ices of water and other substances including ammonia and methane, mixed in with various types of rocks in varying proportions.
And there are a number of ice covered objects in the outer solar system which have been found to have, or may possibly have, dense layers of ice entending down for many kilometers below their surfaces, with oceans of liquids belwow the ice and above the rocky surfaces of those objects.
And of course there is speculation about the possibility of lifeforms in those interior oceans.
If there is liquid water using life in some of those interior oceans, and if the circumstellar habitable zone for liguid water using life is interpreted as including the regions where liquid water using life can exist in interior oceans, the habitable zone for liquid water using life should overlap drasically with the habitable zones for liquid ammonia using life and for liquid methane using life. If such hypothetical biochemestries are possible, of course.
Lifeforms which exist on Earth are highly stratified by altitude. The biosphere of Earth extends kilometers high into the atmosphere and extends kilometers deep under the surface of the oceans, and also kilometers deep below the ground where microbes exist within rocks.
In the oceans of Earth, lifeforms are adapted to a comparatively narrow range of depths and the varying water pressures, temperatures, oxygen levels, etc. at those depths, and will die outside their depth ranges.
So I can believe that possibly some worlds might possibly have interior oceans many times deeper than those of Earth, with lifeforms having totally different biochemestries living at various depths, possibly including layers with liquid water using lifeforms, layers with liquid ammonia using lifeforms, layers with liquid methane using lifeforms, etc.
Such a hypothetical world could be considered to be within several different circumstellar habitable zones at the same time, if the habitable zones are defined as extending to habitabilty for lifeforms in interior oceans as well as for lifeforms on the surface.
As I wrote above, the inner and outer edges of the Sun's circumstellar habitable zone are not known with certainty. but itis certain that the Earth, which has had liquid water using life for at least three billion years, orbiting at a semi/major axis of 1 AU, is with the Sun's circumstellar habitable zone.
So any planet orbiting a star at a distance where it received exactly as much radiation from its star as Earth gets from the Sun, at a distance which I call the Earth Equilvalent Distance or EED, would certainly be within the circumstellar habitable zone of that star.
Here is a link to a question:
The answer by user177107 has a table listing various properties of stars of several different spectral classes and subclasses, including their EEDs. This table will be useful later on.
In Habitable Planets for Man, 1964, Stephen H. Dole discussed the properties of planets habitable for human beings (and thus for beings with similar rquirements), which of course should be a smaller subset of the broader category of planets habitable for liquid water using lifeforms in general.
[to be continued]
On pages 61 to 63 Dole concludes it should take at least 2 billion to 3 billion years for newly formed planet to become habitable for humans. Since the breathable oxygen rich atmosphere necessary for humans is mostly produced by photosynthisis by plants, a planet can be habitable for some forms of life billions of years before it becomes habitable for humans.
Because more massive stars use up their nucler "fuel" much more rapidly, Dole finds an upper limit to the mass of a star which can shine steadly in the main sequence phase of its existence for long enough.
Dole says on page 68 that only main sequence stars less massive than 1.4 solar mass, F2V stars, can remain on the main sequence for up to 3 billions years and have planets habitable for humans. Of course more massive stars could remain on the main sequence long enough for some of their planets to develop lifeforms which don't need oxygen rich atmospheres.
Dole finds a lower mass limit for stars with planets habitable for humans. Because small changes in stellar mass cause much larger changes in stelalr luminosity, the luminosity of low mass stars is very low and their circumsteller habitable zones (which Dole calls "ecospheres") have to be much deeper into their gravity wells. Thus the gravity and tidal forces which low mass stars exert on any planets which might be in their circumstellar habitable zoens will be much stronger.
Strong enough tidal forces will slow down the rotation of a planet until it becomes tidally locked to its star. It's rotation will slow down until its rotation period equals it's orbital period around the star. One side o fheplanet will always face the star and willhave eternal day, and the other side will always face away from the star and have eternal night.
The day star will get very hot and the night side will get very cold. And it is possible that all the air and water on the planet will eventually feeze solid on the night side leaving the planet waterles, airless, and lifeless.
Dole calculated that a star with about 0.88 solar mass would start to have the the inner part of its habitable zone included in the zone of tidally locking. With less massive strs, the zone of tidal locking would fill mor and more of the circumstellar habitable zone, until with stars of 0.72 stellar mass the zone of tidal locking would fill all of the habitable zone.
On pages 71 to 72 Dole says that only stars with more about than 0.88 soar mass canhave full habitable zones, and stars with less than 0.72 solar mass (spetral class K1V) will have no habitable zone left - at least no habitable zone where the planets would not be tidally locked, and being tidally locked which might make them uninhabitable.
According ot the table mentioned above:
A G8V star would have a mass about 0.94 solar mass, a K2V star would have a mass of about 0.78 solar mass, and a K5V star would have a mass of about 0.68 solar mass. The the orbital periods and years of planets in their EED orbits would be 280.06, 182.93, and 114.48 Earth days respecively.
Acccording to this article, a G9V star would have a mass of 0.9 solar mass.
Accoring to this article a K0V star would have a mass of 0.88 solar mass.
And according to this article a K4V star would have mass of 0.73 solar mass.
Anyway, the shortest years of any habitable planets would be in the range e of about 114 to 280 Earth days, unless it is possible for tidally locked planets to be habitable.
Since the majority of stars are class K and class M stars less massive than spectral type K1V, that would mean that the majority of stars are incapable of having human habitable planets.
So the question of whether most K class orange dwarf stars and all M class red dwarf stars can have planets with life is a big question in discussions of life in the universe.
For many years, it was believed that life on such planets would be limited to a ring-like region known as the terminator, where the star would always appear on or close to the horizon.[further explanation needed] It was also believed that efficient heat transfer between the sides of the planet necessitates atmospheric circulation of an atmosphere so thick as to disallow photosynthesis. Due to differential heating, it was argued, a tidally locked planet would experience fierce winds with permanent torrential rain at the point directly facing the local star, the sub-solar point. In the opinion of one author this makes complex life improbable. Plant life would have to adapt to the constant gale, for example by anchoring securely into the soil and sprouting long flexible leaves that do not snap. Animals would rely on infrared vision, as signaling by calls or scents would be difficult over the din of the planet-wide gale. Underwater life would, however, be protected from fierce winds and flares, and vast blooms of black photosynthetic plankton and algae could support the sea life.
In contrast to the previously bleak picture for life, 1997 studies by Robert Haberle and Manoj Joshi of NASA's Ames Research Center in California have shown that a planet's atmosphere (assuming it included greenhouse gases CO2 and H2O) need only be 100 millibar, or 10% of Earth's atmosphere, for the star's heat to be effectively carried to the night side, a figure well within the bounds of photosynthesis. Research two years later by Martin Heath of Greenwich Community College has shown that seawater, too, could effectively circulate without freezing solid if the ocean basins were deep enough to allow free flow beneath the night side's ice cap. Additionally, a 2010 study concluded that Earth-like water worlds tidally locked to their stars would still have temperatures above 240 K (−33 °C) on the night side. Climate models constructed in 2013 indicate that cloud formation on tidally locked planets would minimize the temperature difference between the day and the night side, greatly improving habitability prospects for red dwarf planets.4 Further research, including a consideration of the amount of photosynthetically active radiation, has suggested that tidally locked planets in red dwarf systems might at least be habitable for higher plants.
So it is still an open question whether tidally locked planets can support liquid water using lifeforms.
I our solar system, the giant planets all orbit much farther from the Sun than the sun's habitable zone, and can be called "cold Jupiters".
And above I have mentioned "hot Jupiters", giant planets that orbit very close to their stars and are super hot.
So it is logical to assume that "warm Jupiters" exist, giant planets which orbit within the habitable zones for liquid water usinglife forms of their stars. And a number of such "warm jupiters" have been found.
Probably almost all giant planets have a lot of moons, varying greatly in size, up to in some cases moons that would be large enough to support lifeforms.
Hypothetical giant exommons orbiting around "warm Jupiters" in their star's circumstellar habitable zones could be habitable for liquid water using life. Since most moons would be tidally locked to their planets, the hypothetical giant moons of hypothetical giant planets in the habitable zones of low mass class K and class M stars would be locked to their planets and not to their stars.
So those hypothetical giant exomoons orbiting giant exoplanets in the habitable zones of low mass stars would have rotational periods which were as long as their orbital periods around the planet. Thus they would have alterations of night and day every few Earth hours or Earth days, instead of having eternal day on one side and eternal night on the other side.
So the potential habitability of exomoons in the circumstellar habitable zones of dim K and M type stars has been studied as a factor which might have a big effect on how common worlds with life are.
How long would the days of such habitable exomoons be?
Dole claims on page 60 estimated that if a planet rotated slower than once every 96 hours (4 Earth days), hte alternations of heat and cold would be too much, and the plants owuld not e able to survive thelong lightness nights.
I don't know how accurate that four day limit is, and I suppose that some lifeforms could survive longer days and nights.
"Exomoon habitabitiy Constrained by illumination and tidal heating" Rene Heller and Roy Barnes, Astrobiology, Volume 13,number 1, 2013 discusses many factors relating to potential habitability of exomoons.
In section 2. Habitability of exomoons, they say on page 3:
The synchronized rotation periods of
putative Earth-mass exomoons around giant planets could be in the same range as the orbital periods of the Galilean moons
around Jupiter (1.7d 16.7d) and as Titan’s orbital period around Saturn ("16d) (NASA/JPL planetary satellite ephemerides)4.
So they think that the orbital periods, and thus days, of the hypothetical habitable exomoons might be between 1.7 and 16.7 Earth days.
And that has some relevance to the types of star systems which such hypothetical habitable exomoons could be found in.
The next statement they make is:
The longest possible length of a satellite’s day compatible with Hill stability has been shown to be about P"p/9, P"p being the
planet’s orbital period about the star (Kipping 2009a)
So that means that the orbital period of the planet around the star has to be at least nine times as long as the orbital period of the moon around the planet, for the moon to have an orbit stable in the long term.
So the minimum orbital period for an exoplanet which has a habitable exomoon should be at least about - times 1.7 to 16.7 Earth days, or at least about 15.3 to 150.3 Earth Days.
If an exoplanet with a year 15.3 days long has to orbit in the EED of its star, the star would have to be somewhere between an M5V with an EED of 11.68 Earth days and a M2V with an EED of 36.5 Earth days
If an exoplanet with a year 150.3 days long has to orbit in the EED of its star, the star would have to be somewhere between an K5V with an EED of 114.84 Earth days and a K2V with an EED of 182.93 Earth days.
And of course a planet with a habitable exomoon could have a year much more than 9 times as long as the orbital period of the moon.
This indicates that some but not all sub classes of class M red dwarf stars could have habitable exomoons orbiting giant exoplanets in their habitable zones.
Dole also considered the possibility of a world that would otherwise become tidally locked to its star becoming tidally locked to a companion world instead, whether that companion world was a moon, the other half of a double planet, or a giant planet tha the world was a moon of.
On page 73 Dole estimated that if mid ocean tides grew to more than 20 feet, the tides at the shores would gradually erode away the continents until there was no dry line left and only ocean dwelling life could exist on the planet.
In the case of a world tidally locked to a companion world instead of to its star, it could have short enough days orbiting within the habitable zone of a star much less massive than 0.72 times the mass of the Sun. But Dole calculated that the tidal forces of the star in such a system would raise tides over 20 feet if the star was less massive than 0.35 solar mass.
So if a planet tidally locked to its star can not be habitable, no star less massive than about 0.35 solar mass should be habitable accoding to Dole.
According ot this table:
An M5V star would have a mass of 0.16 solar mass and an orbital period in the EED of 11.68 Earth days, and a M2V star would have a mass of 0.44 solar mass and an orbital period in the EED of 36.51 Earth days.
According to this article:
A M4V star has a mass of 0.23 solar mass, a M3V star has a mass of 0.37 solar mass, and a M2V star has a mass of 0.44 solar mass.
So since red dwarfs are the most common type of stars in the universe, their abiity to support life is important.
If liquid water using life can exist on tidally locked planets, the majority of planets with liquid water using life may orbit red dwarfs.
If liquid water using life can exist on giant exomoons orbiting giant exoplanets in the habitable zones of red dwarf stars, those exommons might be the most numerous and common type of worlds to have liquid water using life.
If liquid ammonia using life and/or liquid methane using life is possible, planets and moons with such lifeforms orbiting red dwarf stars may be the most common types of life bearing worlds in the universe.
And those possiibities do not seem to me to be mutually inconsistent. It is possible that two or three of them could be be real and coexist.