So while horses do certainly cause damage to trails in high use areas, that's not so much the motivation for allowing them but not bicycles.
I love mountain biking and think there absolutely should be places for it to be done, but I also love designated Wilderness Areas and their restrictions and protections.
Where I live, most Wilderness Areas are relatively little used and suffering from post-fire erosion, and so the trails often suffer from overgrowth and fading tread. As a result, horse use is actually quite welcome as it's currently below the threshold where it goes from helping keep the trail beaten in to actively damaging it.
I think banning commercial packers from the Wilderness would go a long way to fixing the problems with horses. Perhaps with some sort of licensing system, mountain bikers could be allowed as well.