The "billions" estimates are probably correct, precisely how many billions is not going to have any significant effect on your plot line. So I will address a different story concern I'd have.
.. would be more related to gravity than atmosphere, temperature, etc; all of these can be overcome with relatively trivial applications of technology.
Sure, but if you are willing to live on a planet without an atmosphere at -200F then you might as well be living in Space!
non-optional Gravity is a detriment, and in space another relatively trivial technology to make it optional is just centrifugal spinning of a habitat; when Earth gravity is necessary for plant growing, or light gravity aids various manufacturing concerns (e.g. 0.2 G keeps everything on tables and floors, and liquids in their vessels, instead of floating about the lab or factory).
Similarly with a crushing atmosphere, or a molten surface.
Which I think kills your story line as implausible; people would not live on planets just for the gravity when they can live in space habitats, with whatever gravity they select, and that is cheaper, safer, and easier than living on a planet. It avoids the gravity well, too: It is easier to relocate, has no weather or natural disasters, planets can be used as shelter from star radiation; solar energy is easier to gather, structures for factories and energy gathering can be flimsy and built with "support" against gravity.
The only plausible reason I can think of, for occupying a planet and requiring close to 1 Earth Gravity, is to farm or ranch on a huge scale, hundreds of times the size of cities; or perhaps mining for the hundreds of minerals that only form over millions of years in the pressures and heats found on a planet.
We can ignore the cost of getting the product out of the gravity well; the important point here is that to be useful as farms or ranches, the planets must be much more like Earth than just having gravity; they must be able to support the kinds of life we find valuable (food plants and food animals, or food producing animals (bees for honey, chickens for eggs, cattle and goats for milk, etc).
Which brings you back to NASA's estimates of maybe a billion. But you can ditch the idea of trivial tech on this scale, so terraforming takes too many years and and your fictional characters discover that upon examination, the number of planets with atmospheres, water, and non-toxic composition suitable for life are rare, closer to 250,000. Then you enter the realm of plausible scarcity.
Alternatively, if you just want lots of habitable planets, presume Nasa's billion is an underestimate, and most planets in the Goldilocks zone capable of holding liquid water end up having it, and oceans, at least plant life and an atmosphere with oxygen. It is entirely plausible that life almost never evolves past that stage, and Earth is unique in having any large multi-cellular animal life. (Not that evolution doesn't work, but it only works by accidental mutation, and the particular accidental mutation that put Earth on the path to developing the first "animal" life may have been a one-in-a-quadrillion six-bank billiard's shot that just hasn't happened anywhere else in this galaxy.)
You can plausibly make terraforming out of reach so livable planets are rare, or make terraforming unnecessary so livable planets are plentiful. The argument that it is easy to live on desolate planets that wouldn't support life makes no sense, when it would be even easier to live in space.