Also, I think it gets lost in physics that theories are models of reality. They are not reality themselves. General relativity is about as good as a model gets. Alongside the standard model, it's one of the most tested models around.
A singularity has a meaning in physics in terms of math, which is dividing by zero or something approaching zero in the denominator. A singularity has a physically modeled aspect in that it's the name assigned to whatever becomes of the matter that gets squished down. A black hole is effectively not a thing in itself but rather an effect of what happens when mass is squeezed beyond all known limits.
What's inside a black hole is not the only thing in the universe we can't see. The actual universe is far bigger than the visible universe, but we can't see outside of the visible universe due to how light works. So it basically doesn't even matter what's going on there. The question is if anything happening at the boundaries can tell us something about that which we can't see. It doesn't make sense to dismiss what we observe on the outside based upon what we literally cannot know of the inside.
And after all, general relativity is a classical theory. Is it really all that "weird"? It all feels somewhat mechanical and natural as you start learning it and turn off your biased intuition.