I took fencing in college, too, and believe it or not, the instructor's name was Pierce. That was at West Virginia University in 1968. I also got to use the schools rifle range during ROTC using .22 Springfield bolt actions.
I'm not sure that having a dull sword while on horseback for safety reasons is logical. If that were so, then what about guns?
I think a lot of the controversy about swords in the military back then came from the fact that swords were commonly worn with civilian dress, at least by certain classes. They were taught by civilian fencing masters to be able to fight other civilians armed the same way, first with rapiers, then with smallswords, which are pretty much just small rapiers. They weren't intended for military combat exactly, even though they were usually carried into combat by officers. Indeed, the US Army NCO sword of around 1840 is a smallsword and not such a bad sword as such but I'm not sure it would stand up to an attack by a soldier armed with a saber of the period, dull or sharp.
Another thing is that a stabbing sword like the NCO sword won't cut worth anything because there is not enough weight in the blade. Yet the blade needs to be as sharp as it can be just so the blade cannot be grabbed by the opponent without danger of losing fingers in the process. By the way, I think slashing is easier than stabbing, even if stabbing works better.
Lest you think this is all pointless because nobody gets into a swordfight anymore (and very few of us in gunfights, for that matter), the emergency room at the local hospital will let you know that knife wounds are still common, unless you live in a nice peaceful town where nobody needs a gun.