Your first question is the problem: they must always be moving towards the center - which implies they must be getting closer to it. Because if they can never a

Your first question is the problem: they must always be moving towards the center - which implies they must be getting closer to it. Because if they can never actually reach the center, then in 4D spacetime they're no longer moving towards it - they'd be traveling parallel with it. Which would imply that singularity isn't a singularity - i.e. the "hole" it makes would in fact be highly curved spacetime in one dimension, but completely flat in another (i.e. it would be a cylinder). Which creates a hole lot of weird infinities in the system: i.e. a black hole becomes a finite bounded volume on the outside, but contains an infinite amount of space on the inside (since no matter how much you distort spacetime, the molecules of whatever falls in can just pull themselves together more tightly provided the distortion doesn't happen too quickly).