I only have a very basic understanding of general relativity, so this might sound stupid. But as I see that gravitational objects orbiting each other, even if one of them is massive, as long as their paths don't lead to collision, they curve back, like a sling shot.
So since singularity is supposed to have zero radius, it seems nothing should actually collide with it. And even if it's non-zero, only near zero, there could still be a possibility that a particle falling (depending on its initial orbit) could eventually just get a sling shot around it, never colliding with it or falling into it.
Does that make sense? Or do all paths really lead directly at the exact the center of the singularity?