Hang on, I was under the impression nobody actually thought singularities existed, only that our current math and physics isn't developed enough to get any reasonable results in such extreme places?
Nobody who's informed believes that the center is infinite, the general public on the other hand has been told the center of a black hole is a point of infinite density by 'experts' for nearing a century. It's the same mistranslation that happens when a lot of science is published in media, it's easy to just say what an expert knows but it's a lot harder to explain it well. The expert knows TONS of nuance on their subject while the rest of us don't have nearly enough time to become as informed as they are.
You are not wrong. So I am not an expert in this area. My understanding is that we get different answers depending on what branch of math/physics you're using. Relativity theory says it is infinite, quantum theory says it isn't. The consensus of most experts in this field is that the center of a black hole isn't infinite, but that's based more on intuition than actual evidence. So the real answer is we don't know, but the educated guess is that it is not infinite.
For the general public, as you call it, a black hole is giant fixed vacuum cleaner in space, and infinite could as well be a chineese word. And it's not a mistranslation, it's that it doesn't matter.
It's not that maths and physics aren't developed enough, it's that we don't know. General relativity is not wrong, at best it is inaccurate, and we have no way to prove it yet.
It's like newton's gravity : it's not wrong, it's inaccurate outside of some conditions.