There's no particular reason to believe the singularity in the math describing black holes has physical existence. Lots of equations have singularities and usually it just means that you're using them outside the domain in which they're defined.

Maybe some heretofore unknown force kicks in and keeps the material inside the black hole from forming a singularity. Maybe they're actually fuzzballs:

https://en.wikipedia.org/wiki/Fuzzball_(string_theory)

You don't really have to have a fully worked-out explanation of why the black hole singularities aren't physical- nobody can see inside the black hole, so all we can say is that the existence of black holes is consistent with there being a physical singularity in the middle. But really anything could be happening in there, we just don't know. The singularity the GR predicts is arguably evidence against GR holding at the center of the black hole; there's very probably something else going on.

This makes no sense. When you apply the mathematics of fluid dynamics to the problem of water leaking through a dam wall, you get singularities.

It makes no sense to say "you either believe the amount of water in a dam wall becomes infinite and the universe collapses in on itself inside a dam, or you have a better theory, what is that theory?" (you would become a famous mathematician if you had that theory by the way).

There are a few. Fuzzballs in string theory. Holographic /Firewall theories etc.