Well, the impact of riding on muddy trails depends a lot on where you live, the soil and other issues. I 'grew up' riding in Western Oregon where the soil is very loamy and has a lot of bio material in it. The impact of riding on mud there wasn't a big deal; the tracks would disappear quickly, resisted compaction and trenching very well due to all the bio material and constant replenishing of the trail tread, and the trails seemed to hold up pretty well. Most riders didn't worry about riding on mud too much; there was even a popular race series near where I lived called "Mud Sweat and Gears."
Moved to Utah where riding on mud is verboten and I thought it was kinda funny until I saw the impacts. Our soil here has a very high clay content, and we get a lot of sun and dry weather that can dry out a trail very quickly. So a bike (or horse or foot or whatever) leaves an imprint on wet ground, and on saturated clay, you'll sink a few inches deep at times. If that print is not buffed out before it dries, it bakes in the sun and becomes incredibly hard -- we're talking a shovel and pick axe to smooth it out, not just a rake. As a result, it is going to be there for a long time. For example, someone rode a horse up a very popular trail when wet a couple years ago, and the prints (meaning 2" deep post holes) were there all summer. Kind of annoying to ride through others ruts and holes for months because they had to ride when wet.
I don't know about the actual environmental impact of riding trails -- I suspect it is actually very little and quite overblown by 'anti's'. But around here, at least, you can screw up a nice trail for the whole season by riding it on a wet day.