Let me first mention that a maximal atlas contains all possible coordinates you can choose on the manifold. None of them need to cover the entire manifold and often none of them actually do (for example, there is no coordinate system that covers the entire Earth at once: you'll always miss at least one point).
Let us now consider the Schwarzschild spacetime. We equip it with a coordinate system that holds for $r > 2 M$, $t \in \mathbb{R}$. We notice the metric still seems reasonable for $r < 2M$, but at $r = 2 M$ some components either vanish or diverge and we can't really use it there. Since we are typically on situations with $r > 2M$, let us start considering only the $r > 2M$ region.
When doing computations on this region, we notice that there are geodesics that go up to $r = 2M$. This happens in infinite static time $t$, but in finite proper time. This prompts us to investigate that region. In order to do so, we can make a change of coordinates to, for example, Kruskal–Szekeres coordinates. We are doing this transformation just in the $r > 2M$ region, where everything works fine.
Now here's the interesting bit: the apparent singularity at $r = 2M$ is no longer appearing on our new set of coordinates. If we compute the geodesic motion, we'll see a body passing through the region which would be previously identified as $r = 2M$ and keeps going. If, out of curiosity, we decide to make a similar coordinate transformation for $r < 2M$, we find out the body has entered that region which previously we did not know if was physical.
Now we notice that although we initially started with only $r > 2M$, we now realize that there was still spacetime beyond that region because geodesics come and go from places we weren't previously describing. Hence, in our new charts, we include these new regions. I like to parallel differential geometry with cartography and imagine this process as an explorer who had a map of some jungle, for example. One day, they go further than the region shown in the map. Of course, nothing changed in the jungle or in the manifold, but now they might need to use different maps to understand this different place.
Being a bit more direct:
what are we doing exactly performing a singular coordinate transformation? Why is this allowed or even encouraged?
You don't necessarily need to think of the coordinate transformation as being singular. You can just make it for $r > 2M$, $\theta \in (0,\pi)$, where it is not singular, and then notice that geodesics will still go through the region you thought was singular, so you just update your new coordinates to cover that region. F
For example, pick spherical coordinates. $\theta = 0$ and $\theta = \pi$ are singular. Yet, you can perform the coordinate transformation to Cartesian with $\theta \in (0,\pi)$ and afterwards realize that you have geodesics going through $x^2 + y^2 = 0$, so you simply admit $x = 0$ and $y = 0$ as a possible value for your coordinates. In the case for Schwarzschild, we did that, and also noticed this ended up allowing us to cover the region with $r < 2M$ as well in a single attempt.
I think the original smooth manifold views the Schwarzschild coordinate a legit member of its atlas. If we now use these other new coordinates (that please the metric and scientists) to build an atlas, are we getting the original smooth manifold?
As I mentioned in the beginning, a maximal atlas consists of all possible coordinate systems that allow for smooth transitions among themselves. We typically choose to assume maximal atlases out of convenience. When you change coordinates, you don't change the atlas nor the manifold. In a more cartographic language, the Earth won't change if you one day decide you want to use a Gall–Peters projection instead of a Mercator projection. The original choice of coordinates was made out of pure convenience and is not an intrinsic property of the manifold.
There is a minor caveat, however.
Maximal Extensions
I initially asked for us to consider spacetime only for $r > 2M$, as that was the available region for our coordinates which we considered physical. After performing a coordinate transformation, we realized there were other regions our geodesics could access and we chose to consider them as part of spacetime as well. In a more technical jargon, we found an extension of the spacetime. In the particular case of the Schwarzschild solution, we found the unique maximal analytic extension of the Schwarzschild spacetime. In other words, the unique analytic manifold that coincides with the Schwarzschild metric in the region $r > 2M$ and cannot be further enlarged.
The thing is: initially, if we so desired, we could consider the original spacetime to be only the region for $r > 2M$ (i.e., we could consider spacetime to be only the right wedge of the Schwarzschild spacetime's Penrose diagram). In this situation, once we changed coordinates and allowed them to run over to values different than $r > 2M$, we ended up in a larger manifold (larger in the sense it contains the original spacetime as a submanifold).
Typically, we assume all spacetimes to be maximally extended, i.e., we assume that it is impossible to find another spacetime that is larger than the one we're considering (see Hawking & Ellis 1973, Sec. 3.1, for some more detail on the technical bits). From a physical point of view, this means demanding from the start that if a geodesic can reach some point, then that point should belong to spacetime. It is the view that we screwed up by taking the Schwarzschild coordinates too seriously: they are just an arbitrary choice of coordinates and never had the obligation of covering up the whole manifold. It is the view that we should recognize the poles are a part of Earth, even if their longitude is ill-defined. The view that the jungle exists regardless of whether our maps cover it.
