I haven't seen anything to fully convince me that mules are more sure-footed than a horse. I am convinced that mules tend to "think" a little more than a horse, but a horse that has been through that kind of country and been exposed to that kind of stuff can handle it pretty well.
Until you actually do some of that kind of riding, you simply have no concept of what a horse can really do. Problem with horses is that they panic when they get into a tight situation they can't see their way out of (a lot of that has to do with the rider, as well, in my opinion). Mules tend to stop and think and won't move until they see their way out. After a horse has been exposed to that kind of terrain and has had a rider who understands what a horse can and cannot do, they settle down and pick their steps very well.
I have also found that riding a horse barefoot on the trail during training teaches the horse to watch where it puts its feet. My mare will now watch where she puts every hoof in any king of rough terrain. She will walk through rocks, deadfalls, over logs, etc, watching very carefully where she places her hooves.
As for harm to the horses, in my experience, you see far fewer serious injuries to man/woman and beast doing stuff like this than you will in the show circuit performance competitions, western or English.
There used to be a video on the Extreme Mules youtube channel where they took horses along for the ride, doing all the same things. You can also look at the channel for Extreme Outlaw Rides. He does all that stuff with mustangs.