Here's a good playlist to warp your head around the subject.
https://youtube.com/playlist?list=PLsPUh22kYmNBl4h0i4mI5zDfl...
Specifically, what happens to fermions once all the quantum states are filled and they are still constrained by all the other fermions around them? We have no idea (and everything seems to point to a collapse into a singularity).
If, instead, there is an actual singularity (which has infinite density) it means that the curvature of space time is infinite, which our current theories can't cope with. Additionally, if singularities have infinite density, how is it that black holes can have different masses? A singularity can only be characterized by its position in spacetime since it has no size and so there is no space/surface for it to have any other property and yet we see that there are black holes with different masses. Another thing we can't explain with our theories.
So yeah, black holes mess with our theories in a fundamental way because as soon as you start pondering what happens inside them we discover that general relativity or quantum mechanics or both break down and so they must be incomplete. Spacetime, being a construct of general relativity is therefore also an incomplete description of the real fabric of space time.
Because if we just accept that, what do we do then? We just sit and wait for some new astronomical observation to give us a clue? that could take forever, and we'd be banking that we have the tech to observe this magic hint.
Better to suppose that the theories we have comprise an accurate approximation or partial model of reality, and from there strive to find a better model.
The process of doing so will either refine/entrench our current model and our conviction, or it will result in actually finding a better model. Win-win, and all the while, we can still have our telescopes and detectors on for the magic hint we'd be sat waiting for anyway.
When things break down into singularities, it can be a pretty good indicator that we've got something wrong. Not necessarily, but in this case, I think we missed something.
For it to just be a super dense “new class of star” would first require you to explain why general relativity is completely wrong.
And that is the problem. Black holes are weird because they break the well tested equations we currently use to describe what we observe in the universe.
If you want my opinion, figuring this shit out (including what we are calling “dark matter”) is gonna unlock a whole new realm of cool stuff for humanity. I suspect there is a reason why we haven’t solved the Fermi paradox and it is because most “intelligent life”, as we imagine it, is living outside our current understanding of the universe. To get into the “cool aliens club”, our understanding of the universe will need to change.