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?
Not really sure about it, but as I understand it, affine geometry can be translated as linear geometry. Think euclid geometry in a 2d plane. In this black hole, I think it means that a particule behind the event horizon will go straight to the center. So every trajectory behind the hoziron is a line that goes to the center, and this center is the singularity.