Why, at the center of a black hole, does the need to be a point where space-time curvature is infinite? I understand that black holes scale with consumed matter, and that the point has 0 density but how do we get infinite G? I'm just an amateur but because this might just be a mathematical implication, and I'd love it if you threw an equation for me to try and pick apart. Also, why wouldn't a model without a singularity work?
The singularity is a result of the way the equations work out in General Relativity. Whether a singularity exists in real life is unknown. A number of people believe that the singularity is simply an artifact of our incomplete knowledge of physics at that scale and that a quantum theory of gravity would solve this problem. Put simply, a singularity doesn't have to exist. It may or may not exist in reality.
There is a very similar thread about this, only a few threads down: https://www.physicsforums.com/showthread.php?t=758146