Homogeneity of the early universe by itself is not enough to prevent collapse. You need rapid expansion as well. A homogeneous universe that is not expanding at all will collapse.
You are answering a different question. The collapse of a homogeneous (i.e. FRW) universe produces an increasingly dense, but still homogeneous universe. Without a central singularity and an outside, it looks nothing like a black hole (as asked).
BTW, expansion alone is not quite enough to answer why the early universe didn't collapse. You can have very rapid expansion, yet a closed universe which eventually comes to a standstill and then starts shrinking. A better answer is that the universe is very close to having critical density, i.e. just enough mass to expand forever at an asymptotically declining rate, absent new drivers of expansion (i.e. dark energy).
One thing our answers have in common is fine-tuning: the universe started out ridiculously homogeneous, and ridiculously close to the critical density. Inflation provides a way to explain both those properties, but (as critics are fond of pointing out) at the cost of fine-tuning the hypothetical microphysics needed to drive it.
