The horizon problem seems to ignore one thing
The big bang was happening 'in parallel' throughout the young universe. If you want consistency between one end of the universe and the other, the light does not need to traverse the entire distance to create adequate influence and uniformity -
For example, if light could only reach 25% of the way across the universe, then every point in the universe could influence a large part that almost totally *overlapped* its neighbour.
As a result there would be a pressure to achieve local uniformity where local was actually something that was interlinked across the whole universe.
Anyway, my feeling (and no, I don't profess to know) is that you don't need to address the horizon problem to explain overall uniformity in a system, particularly where that system is undergoing the same basic process throughout.