Not an expert; please correct me if anything is wrong.

My understanding is:

1. there needs to be somewhere to escape to, for the schwarzschild radius to make sense, and there's no "outside" to the universe

2. we don't know how big the universe is or what it looks like; if it's infinite and uniform then there won't be any central point for a singularity

3. as others have mentioned, inflation is a get-out-of-black-hole card

4. the physics of the early universe aren't necessarily the same as physics now