Why singularity in a black hole, and not just "very dense"?

It's important to understand the context in which statements like "there must be a singularity in a black hole" are made. This context is provided by the model used to derive the results. In this case, it was classical (meaning "non quantum") general relativity theory that was used to predict the existence of singularities in spacetime. Hawking and Penrose proved that, under certain reasonable assumptions, there would be curves in spacetime that represented the paths of bodies freely falling under gravity that just "came to an end". For these curves, spacetime behaved like it had a boundary or an "edge". This was the singularity the theory predicted. The results were proved rigorously mathematically, using certain properties of differential equations and topology.

Now in this framework, spacetime is assumed to be smooth - it's a manifold - it doesn't have any granularity or minimum length. As soon as you start to include the possibilities of granular spacetime, you've moved outside the framework for which the original Hawking Penrose theorems apply, and you have to come up with new proofs for or against the existence of singularities.

Because otherwise general relativity would contradict itself. The event horizon of a black hole is where not even light can escape. Below the horizon all photons must fall. In relativity theory all observers measure the speed of light the same, c; that's a postulate of the theory. Then all physical things (including observers) at and below the horizon must fall and keep falling, lest they measure the speed of light emitted upward to be something other than c. If you could stand on a very dense lump of matter of finite size at the center of a black hole, and pointed a flashlight upward, the photons would somehow have to fall to the ground (without moving upward at all) and you wouldn't measure the speed of light to be c in the upward direction. The theory would be broken. The singularity is the "can't fall further" point and the theory becomes inapplicable there.

See Carter 1968 for why rotating black holes that have incoming disturbances may not have a singularity at all.

A stationary non - rotating hole will have a singularity. But no one thinks that these exist in nature. But with rotation that singularity 'shrinks' to a ring. The set of paths that hit the singularity is shrunk to a mathematical 2D plane from 'all directions' with the Swarzschild Soln. Then with incoming 'noise' it may be that there are no paths - geodesics - that lead to a singularity.

http://luth.obspm.fr/~luthier/carter/trav/Carter68.pdf

All exact solutions of General Relativity are done with asymptotically flat space, which does not exist in the real world. So while the theory of GR admits singularities, in a real classical GR world they likely don't exist.

Carter actually always talks about a singularity, but one with no paths to it. No ouchy at the end of a path. With no paths to a singularity - is it really there? I would think not, and as Carter points out, others do too. (Lifshitz and Khalatnikov).