I don't think "gaiting horse" and "jumping horse" are mutually exclusive.
Gaited horses can jump just as well as any other "non-jumping" breed. Whether they make terrible/good/excellent jumping horses is all about the training that goes into their jumping and their ability, not their breed or gait.
I gait my horse up to jumps sometimes. She very well can jump from a gait, but it isn't in good form and I wouldn't do it approching a high or difficult fence.
I taught her to trot. It's pretty easy to do. Even the most lateral horse will pick up a trot over poles. It's all about having a clear trot cue versus gait cue when you take the poles away. (I go into two point and encourage her to lower her head and round her back.) Sometimes my horse will gait to the first jump, get the memo, and then trot the rest. They'll do what's comfortable.
You can also canter jumps. Gaited horses can canter. I don't know if yours does; some people don't teach them. I know some people don't have rackers canter; I don't know anything about paso training standards.
In my experience, gaited horses are just as athletic and able to do performance events as any. I wish I had some video of my mare jumping a little course. She looks just like a green jumper. Not a gaited green jumper, but just a green jumper.