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.
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.
A singularity is the single point mass at the center of an ideal (Schwarzschild) black hole. But mathematically, that can only happen if the mass that forms the black hole isn't rotating. In reality, all the mass in the universe is moving around, because mass is not distributed uniformly, so gravity is pulling stuff around in a big mess. So when a black hole forms, it's definitely a rotating (Kerr) black hole.
A rotating mass has different gravity than a non-rotating mass. Not by much, but when you've got the enormous mass of a black hole, it becomes significant. This causes objects "falling into" a black hole to "miss" the point at the center, and form more of a cloud during spaghettification.
The article is fairly accessible if you sit down and read it.
Honestly, inside the event horizon, everything stops making sense compared to our day-to-day experiences. The immense gravitational forces distort space and time. It doesn't really make sense to think about objects remaining intact as recognizable objects once they cross the event horizon.
I read it and understood the setup in the first section that you're paraphrasing about ideal black holes with the 1963 kerr advance and the orbits, but don't know if I understood the whole reasoning behind the thesis of the article in the later sections of the article, that singularities don't exist.
If you scroll past the opening explanations of ideal black holes,
Kerr later asserts that because of the elliptical orbits of matter and light trapped inside the inner event horizon due to black hole rotation, there's no central singularity and just an inner cloud of matter perpetually traveling in elliptical orbits. I think this is the spaghetti you're talking about.
Is that assertion of the structural difference in practical black holes the whole point of the article?