There are two separate issues here.
The Schwarzschild radius for a black hole, is calculated based on some very specific assumptions. For example, they won't be valid when space is rapidly expanding.
They also may not be valid on a huge scale, such as galactic clusters, where "dark energy" or expansion are factors, but we don't know enough about those to be sure of all their effects. But intuitively, it seems likely that our usual equations for collapsing objects wouldn't apply (or would require major modifications) if we can't assume a locally uniform spacetime metric, so they would apply on small scales up to local galaxies but probably not to large regions of space where expansion varies, or times of extreme expansion, or to the universe as a whole.
That's why the initial universe, though very dense, didn't re-collapse. The equations that determine when collapse happens, which we can apply in the current universe, are based on assumptions and approximations that just wouldn't be valid in the conditions of the early universe.
The other issue going on, is the observability and horizon of our universe, which is for a completely different reason. Special relativity is the principle / natural law which says that nothing can travel (no known type of signal can propagate) faster than the speed of light. But special relativity applies to propagation within spacetime. In the very early universe (and much less nowadays) spacetime itself was expanding. This wasn't an expansion like we are used to. It was a change to the actual geometry of spacetime itself. As such, it didn't have a limit to its speed. It happened a tiny fraction of a second after the Big Bang. ** Suddenly, points in space that were "close together" (in some sense) became trillions of trillions of times more distant in a very short time. Wikipedia says the expansion was of the order of 10^26 in linear dimensions, or 10^78 in volume terms.
In "intuitive" rather than precise scientific terms, points that might have been reachable by light from each other in moments suddenly found themselves so distant that light needed immense amounts of time to travel between them.
If two such points suddenly found themselves much more than 13.7 billion light years apart (due to expansion), then there wouldn't have been time for light from one point to reach the other, even in the 13.7 billion years since that huge expansion. So they literally would not be observable now, because signals couldn't reach us in any way from them. ** Hence this means there's a practical "radius" or limit to what we can hope to observe, set by the speed of light itself - called the observable universe.
** We could in theory observe some of these distant objects from times before that expansion, when they weren't located outside the observable universe, but the expansion occurred in the first 10^-32 or so of a second, when the universe was so energy intense that we can't really hope to ever observe anything from that era.