Very briefly visiting... hopefully back properly soon.
When too much matter is gathered in too small a space, then you get a black hole.
Sometimes, not always. To get a black hole you need to have sufficient matter concentrated within a volume, surrounded by space of much lower density, usually vacuum. Even then, there are several requirements. This is not the situation with the early universe. There you have uniform density of matter throughout space. These two situations give rise to wholly different space-time solutions.
As Chirop very rightly said in answer to the original question (what force was powerful to overtake all that gravity in order to get the universe to spread out?) it's just General Relativity. GR is non-linear. That means that you cannot extrapolate one situation and hope to get anything resembling some other situation. That's what makes exploring GR so exciting... you never know what you are going to find.
Also, be careful of thinking too three-dimensionally. GR is a 4d theory and only really makes sense when viewed in 4d, especially the dynamics of the big bang/early universe. Our everyday idea of gravitation as gravity - the force between massive objects - does not extrapolate well up to the universe as a whole.
I have a feeling the explanation is going to involve inflation
No, not at all. Inflation is merely there as a suggested way of explaining some residual oddities. The Big Bang is entirely GR where-as inflation is GR+QFT, venturing into semi-classical gravity.
There was no solid matter to spin at the time of the big bang
Just as an aside, you don't need solid matter to have angular momentum. Space-time itself can posses ang mom. The Kerr solution is an entire universe that spins, but has no mass. Very non-Machian.
Will keep an eye out for any replies but rapid responses can't be promised... sorry.