Once again, I think @Scribbler makes a very valid point about riding traditions in America.
As one who had my earliest lessons from one of those cavalry officers, and who has since ditched virtually everything he and his riding school taught in favor of much more updated and effective riding techniques, I can say that it was mostly staying on the horse and basic commands for steering and gait.
That said, the very small handful of us who continued on got a good enough grounding that we were considered some of the better riders in the area. That didn’t mean that we had a speck of riding sophistication. We were just safe enough and manageable enough on a horse that we could jump a 3’6" ring course (something almost everyone who rode more than 3 years could do then), survive some degree of bad horse behavior, and ride safely in open country. We were not on any trajectory to grander things in our riding careers.
We were “better” not because we were sophisticated riders (we weren’t), but because we had learned on a motley collection of horses of unknown background. Not on current or former show horses, or even horses that had a nice trail-riding past. We had no qualms riding any horse and could manage most. Our compatriots who did learn to ride in more sophisticated show barns didn’t have as broad an exposure to un-schooled horse behavior.
Honestly the former-cavalry riding school was intended to be a starting point for older children (youngest age they would take was 12), not a course for finishing riders. The assumption was that as the learners put in two or three years in the riding school, acquiring some basic skills (progress tended to be slow), either their parents would move on to a more show-oriented training barn, or perhaps the kids would quit riding for other interests as they got older, but later provide their own children with riding lessons. The real point was simply to keep riding alive in the gentrifying community.
This instructor gave us access to some of the background material for the cavalry teaching. Apparently, maybe 100+ years previously, many new cavalrymen were city boys who had never been around horses, but who thought that the cavalry would be an easy ticket out of the infantry. And apparently they were accepted, because the cavalry needed the human numbers.
And further, the horses had been haphazardly gathered – almost confiscated, really, by gov’t order with minimal reimbursement to their previous owners – from farms (not raised in a cavalry system). The selected horses were of varying ages, genders and training backgrounds, and could not be assumed to be reliably rideable at the beginning of training. Some were, but not all. (There is a whole side rabbit-hole of extensive reports and complaints from the regular cavalry about the inadequacies of horse selection and selectors up through WWI, if you know where to find them.)
So the teaching method was mostly how to keep these new cavalrymen in the saddle and give them the most basic skills for controlling horses, sometimes with large, harsh bits, and even harsh spurs. And hope that the riders and the horses would figure some things out and make peace with each other.
Anyway. Yet another point in American riding where teaching across large parts of the country was not terribly systematic or goal-oriented beyond a basic ability to stay on a horse and direct it. And had no higher ambitions than producing riders who were adequate enough for pleasure riding, and perhaps maintaining a few backyard horses.