**sigh** Every breed of horse can jump...something. Every breed of horse will trot. (I guess some of you don't have your gaited horses in your back yards to watch them. I do and they DO trot...often.) Every 1/2 bred gaited horse wants to gait.
I have 2 gaited and one non-gaited--QH. The QH is the one I will train to jump bc his build is better for jumping. If you want a hunter or a jumper you will NOT pick a gaited horse.
The US Cavalry disbanded (in 1942) bc of machinery AND bc they tried for decades to create the "superhorse" that could do it all, but that isn't possible.
You can enjoy popping over 2 ft. obstacles with your gaited horse, but you're never going to be able to really compete in the show ring with horses that are natural jumpers. The only REALLY great jumping horses I've owned were a QH and an OTTB. Both of them smoothly "took you with them" when they went over a jump. ALL of the others would pop high and it was difficult for me to land the jump.
The U.S. Horse Cavalry was actually disbanded in 1948. Training at Ft. Riley continued through at least early 1944. The last formal units (other than ceremonial mounted color guards) were the Mounted Constabulary in Germany. They functioned until the early '50s.
The Remount Service did, in fact, produce a very good, very consistent horse through their program of "public stallions." It was good enough that by the 1930s the production was so consistent that the Phillips Officers Saddle was produced in only one tree size. I'm not sure that there was more than one seat size. These horses were remarkably well suited to duties that the Cavalry had assumed by that time. Of course mechanized, motorized, and armored forces, as well as aircraft, were causing rapid obsolesence for the horse mounted trooper.
In Europe the Germans and Russians would maintain divisional sized mounted forces until the end of the War. The conditions in the East favored the mounted soldier for many missions.
Today at least China and Chile maintain significant mounted arms.
I completely concur that most gaited horses will be marginal jumpers under most conditions. This does not mean that you might not find the occational excellent jumper, but that horse will have a more diagonal gait which will permit more impulsion from the rear. Laterally gaited horses generally lack this level of impulsion.
As an aside, very few armed forces ever used gaited horses in numbers. The only one that I know of that ever did was the CSA, and that's because each trooper provided his own horse and gaited horses were very common in the South.