You hear statements like "physics breaks down at the singularity of a black hole".

Physics (the real Physics) doesn't break down ever, it's your theory that breaks down, it's your theory that is wrong.

General relativity gets a lot of things right, it seems to explain what happens outside of a black hole well. In the usual approach to the Schwarzchild solution (a black hole with no spin and no electric charge) you usually start in a coordinate system that has a mathematical singularity at the event horizon then make a coordinate change that makes that singularity go away (with the deceptive feature that the event horizon seems to be a global instead of local property of the black hole.)

My belief is that in quantum gravity that coordinate translation isn't valid so the classical picture of the black hole interior is "not even wrong". It drives me nuts that every week a new QG paper comes out that the media describe as "a resolution of the Hawking paradox" despite the fact that Hawking admitted he was wrong about the paradox and there is no paradox a long time ago.