Inflationary theory. Someone more educated on the topic can pine in, but my lay understanding is that the output of some form of primeval radioactive decay (Big Bang) caused a phenomenal rate of spacial expansion, which caused space to expand faster than the rate of gravitational collapse in those first moments.

It's not the rate of gravitational collapse you have to beat, it's the speed of light.

Although I guess that makes sense. After all it's spacetime, not space. So if the expansion rate is higher than the speed of light, then you don't have the black hole problem.

The universe is and always was infinite even at the big bang. At the big bang although the universe was incredibly dense it was so uniform there was no net direction of gravity to cause clumping. That and inflation rapidly expanding away any quantum imperfections beyond causal contact.

Why does it avoid the problem?

> Over billions of years a contracting scale factor brings everything a bit closer, but not all the way down to a point. The dramatic change comes from the Hubble radius, which rushes in and eventually becomes microscopic

All the matter of universe at a microscopic radius? That doesn't avoid the problem.

There has to be spacetime to form a black hole, where would it find such spacetime before the big bang...

Well what about 0.000001 seconds after the big bang (add zeros as needed).