In most texts on GR we are first introduced to a formal and rigorous definition of a manifold.
We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. The Einstein equation relating curvature and energy is invariant regardless because of tensor transformation law. Put it another way, we are not limited to only Lorentz linear transformations as in SR, hence GR is indeed the general theory of relativity.
But I have not seen examples in GR texts in which the rigorous definition of the manifold, i. e., that there must be smooth charts in an atlas that covers the entire manifold is used explicitly every time we introduce a new coordinate system.
The question is when and where one needs to check this out and does not count on an intuitive sense?
Or can you suggest an example in which we build a new arbitrary coordinate system and the entire manifold cannot be covered through a maximal atlas?
Should we be alarmed about the violation of the manifold's definition only where we see metric singularities in all possible candidate coordinate systems, e.g. at Big Bang or that inside a black hole?