This is a difficult problem for those the least concerned with scientific plausibility. At the end of a long discussion I come up with a reasonably plausible planetary set up - though it is so unlikely to occur naturally that a writer might want to have characters speculate that an advanced civilization actually arranged the planets that way.
PART ONE: A RING OF PLANETS
This looks like a job for Sean Raymond of the PlanetPlanet blog.
In this post: https://planetplanet.net/2017/05/03/the-ultimate-engineered-solar-system/1
Raymond mentions a paper that shows that it is possible for several astronomical bodies to share the same orbit around another body under some circumstances.
Orbital stability of systems of closely-spaced planets, II: configurations with coorbital planets Smith, Andrew W.; Lissauer, Jack J.
Celestial Mechanics and Dynamical Astronomy, Volume 107, Issue 4, pp.487-500
Their calculations indicate that at least seven bodies of nearly identical mass would have to be spaced evenly along the orbit for it to be stable. So your requirement for ten would be quite possible.
If there are ten equally spaced co orbital astronomical bodies they would be spaced 36 degrees apart along the orbit.
If those planets orbit their star at a distance of 100 units, the total circumference of the orbit will be about 628.318 units. One tenth of that will be about 62.8318 units. How close can the next planetary orbit in or out be?
According to this Wikipedia list:
the smallest known ratio between the semi-major axis of two consecutive planetary orbits is about 11 percent, with Kepler-36b & Kepler-36c.
So if that was close to physical lower limit of ratio the next closest inner planetary orbit would be about 90 units or fewer from the star in the system, and the next closest outer planetary orbit would be about 111 units from the star.
Thus your planet could get within ten or eleven units of a planet in the ring of planets when it passes closest to it, which would be about six times as close as the next two closest planets in the ring will be at that time.
Note that the ratio between orbits is different from the absolute difference between orbits. The smallest known difference between two consecutive planetary orbits is in the Kepler-70 system. Kepler-70c is believed to orbit only about 0.0016 Astronomical Units, or about 240,000 kilometers, beyond the orbit of Kepler-70b.
During closest approach, Kepler-70c would appear 5 times the size of the Moon in Kepler-70b's sky.
So it is possible for two planets to appear as orbs in each other's sky at their closest approach while remaining mere dots in the sky at other times.
And it is suspected, but not proved, that a third planet might orbit between Kepler-70b and Kepler-70c, which would sometimes get an ever better view of Kepler-70c.
Of course the planets in the Kepler-70 system don't orbit in the habitable zone of Kepler-70. But TRAPPIST-1 does have planets in its habitable zone that do orbit very close to each other.
This work used the Spitzer Space Telescope and the Very Large Telescope at Paranal, amongst others, and brought the total number of planets to seven, of which three are considered to be within its habitable zone. The others could also be habitable as they may possess liquid water somewhere on their surface. Depending on definition, up to six could be in the optimistic habitable zone (c, d, e, f, g, h), with estimated equilibrium temperatures of 170 to 330 K (−103 to 57 °C; −154 to 134 °F).5 In November 2018, researchers determined that planet e is the most likely Earth-like ocean world and "would be an excellent choice for further study with habitability in mind."
The orbits of the TRAPPIST-1 planetary system are very flat and compact. All seven of TRAPPIST-1's planets orbit much closer than Mercury orbits the Sun. Except for b, they orbit farther than the Galilean satellites do around Jupiter, but closer than most of the other moons of Jupiter. The distance between the orbits of b and c is only 1.6 times the distance between the Earth and the Moon. The planets should appear prominently in each other's skies, in some cases appearing several times larger than the Moon appears from Earth. A year on the closest planet passes in only 1.5 Earth days, while the seventh planet's year passes in only 18.8 days.
So the TRAPPIST-1 system is an example of a star system where planets in the habitable zone usually look like dots in the sky but sometimes pass close enough to appear as visible orbs and even appear as large as the Moon in Earth's sky or even larger.
Calculations of how close planetary orbits can be are usually based on the assumption that all planets in a system orbit their star in the same direction, which is the direction that the star rotates in. In fact, in our solar system all the planets orbit in the same direction, and they all (except for Venus & Uranus) rotate in the same direction. And many of the moons also orbit in that same direction.
The majority of discovered exoplanets orbit in the same directions as their stars rotate in, and in the same directions as any other discovered exoplanets in their systems.
So it is a safe assumption that in the majority of star systems all planets orbit in the same direction.
But how would it affect orbital stability if the planets in a star system orbited in two different direction, perhaps even alternating the orbital direction from one orbit to the next?
Sean Raymond in this post:
discusses a paper calculating the stability of planetary orbits:
Orbital stability of systems of closely-spaced planets,
Smith, Andrew W.; Lissauer, Jack J., Icarus, Volume 201, Issue 1, p. 381-394.
Raymond mentions that Smith and Lissauer show that planetary orbits can be closer together if the orbits alternate between prograde (orbiting in the same direction as the star rotates) and retrograde (orbiting in the opposite direction).
So if the habitable planet in your story and the ring of ten planets orbit in different directions, the planet's orbit can be much closer to that of the orbital ring of planets and during the brief periods when the planet is closest to one of the planets in the ring the other planet can appear much larger in the sky.
PART TWO: A BIG PROBLEM
As planets orbit around their star, their orbital periods are different. The inner planets complete their orbits much faster than the outer planets do. Thee are two reasons for that.
1) The orbits of inner planets are much smaller than those of outer planets, so if they traveled at the same speeds the inner planets would complete their orbits sooner.
2) The necessary orbital speed is faster the closer to the star, so the inner planets have to travel much faster in their orbits.
The combination of those two factors means that inner planets travel more degrees of their orbit per day than outer planets do, and so complete their orbits before outer planets do.
So imagine that an inner planet and an outer planet happen to be lined up with the star in their system. Then as time passes, the inner planet will pull ahead of the outer planet and leave it farther and farther behind. Eventually the inner planet and the outer planet will be on opposite sides of their star. Then the inner planet will begin to catch up with the other planet.
When the inner planet catches up with the outer planet again, the time elapsed will not be exactly one year of either the inner planet or of the outer planet.
The time period for two planets to be in the same configuration with each other and their star is called the synodic period.
For planets outside the orbit of Earth, the synodic period relative to Earth gets shorter and shorter the father they are from the Sun, because they are travelling slower and slower in their obits, so once Earth completes a full circle around the Sun it has to travel shorter and shorter further distances to catch up with the slower moving planets.
Mars has a synodic period of 2.135 Earth years, the asteroid Ceres has a synodic period of 1.278 Earth years, Jupiter has a synodic period of 1.092 Earth years, Saturn has a synodic period of 1.035 Earth years, and so on.
So if the star is like the Sun, and if your main planet is at the distance of Earth, and the ring of planets is at the distance of Saturn, a tenth of the main planet's year would be about 0.100 Earth year or about 36.525 Earth days. A tenth of the synodic period of the ring of planets would 0.1035 Earth years or about 37.803375 Earth days.
So the planet would have same same problem basing their "months" on passing successive planets in the ring of planets that Earth has basing it's months on lunations, the periods between the moon having the same phase. The inner planet may pass all ten planets in the outer ring of planets in most years, but in some years it will pass nine planets or eleven planets.
To make the problem smaller the ring of planets can be farther out compared to the orbit of the main planet in your story.
The semi-major axis of the orbit of Mars is 1.5273 times that of Earth, and the synodic period is 2.136 Earth years.
The semi-major axis of the orbit of Ceres is 2.76596 times that of Earth, and the synodic period is 1.278 Earth years.
The semi-major axis of the orbit of Jupiter is 5.2028 times that of Earth, and the synodic period is 1.092 Earth years.
The semi-major axis of the orbit of Saturn is 9.5388 times that of Earth, and the synodic period is 1.035 Earth years.
The semi-major axis of the orbit of Uranus is 19.1914 times that of Earth, and the synodic period is 1.012 Earth years.
The semi-major axis of the orbit of Neptune is 30.0611 times that of Earth, and the synodic period is 1.006 Earth years.
The semi-major axis of the orbit of Pluto is 39.5294 times that of Earth, and the synodic period is 1.004 Earth years.
The semi-major axis of the orbit of 50000 Quaoar is 43.6916 times that of Earth, and the synodic period is 1.003 Earth years.
The semi-major axis of the orbit of 136199 Eris is 67.6681 times that of Earth, and the synodic period is 1.002 Earth years.
The semi-major axis of the orbit of 90377 Sedna is 506.8 times that of Earth, and the synodic period is 1.0001 Earth years.
Of course Saturn is the farthest naked eye planet known since ancient times. Uranus is sometimes visible to the naked eye, but was never recognized as a planet until 1781. A planet as large as Jupiter or Saturn could be recognized as a planet out to the orbits of Uranus and Neptune.
Of course planets as distant as Neptune, or even Mars, could never appear as discs as seen from Earth.
Any outer ring of ten planets that was distant enough from the main planet in the story to avoid messing up the calendar of the planet more than Earth's is messed up with months that don't fit evenly into years, would many times too distant for the planets in the ring to ever appear as orbs as seen from the main planet in the story. And at any one time most of the ten planets in the ring should be visible from the inner planet in the story, though at any moment only one of them can be close to the position opposite to the star.
PART THREE: AN INNER RING OF PLANETS
What about planets that orbit interior to the main planet in the story?
In our solar system there are two planets orbiting inferior to Earth.
The semimajor axis of Mercury's orbit is 0.3871 that of Earth's orbit, and the synodic period of Mercury is 0.317 Earth years or 155.88 Earth days.
The semimajor axis of Venus's orbit is 0.7233 that of Earth's orbit, and the synodic period of Venus is 1.559 Earth years or 583.9 Earth days.
And I note that the synodic period of Mercury is a fraction of an Earth years, while the synodic period of Venus is more than one Earth year. This leads me to suspect that there is a possible orbit somewhere between those of Mercury and Venus where a planet would have a synodic period equal to an Earth year.
According to my rough calculations, if a planet orbits with an orbital period of about 182.62505 Earth days, it should have a synodic period of about 365.000 Earth days, slightly shorter than an Earth year of 365.25 Earth days which should be close enough.
That calculation should be checked by someone or some program that is better with orbital calculations, and capable of calculating the semimajor axis of the orbit a planet with an orbital period of 182.62505 Earth days.
At such a relative distance compared to the habitable main planet, the ten planets in the ring of inner planets should be hellishly hot and uninhabitable for the natives of the main planet in the story. However, if the main planet in the story orbits toward the outer edge of the star's habitable zone, the inner ring of ten planets might be within the habitable zone and some or all of the planets might be habitable.
Now the problem would be how to get the planets in the inner ring visible only part of the time, so that only one of them at a time is visible from the outer planet.
The planet Mercury can be observed with the naked eye only for relatively short periods when it is separated from the Sun by the largest angles, and only in twilight skies before the Sun rises or after the Sun sets.
So if the ring of planets orbited at the same relative distance to the main planet in the story as Mercury does to Earth, it would be quite likely that only one of the ten planets in the ring would be visible at anyone time.
But unfortunately the ring of ten planets would be at a somewhat greater relative distance and so more than one of those planets should be visible at any one time. Therefore it would be a good idea to somehow make the day sky, the twilight sky, and maybe even the night sky if possible somewhat brighter than the skie sof of Earth, drowning out the light of the inner planets except when they are at extreme maximum elongation from the star in the system, thus making only one planet visible at any one time.