Our riding center received two truck loads, 14-15 feral horses every late June, just caught and started them and had them ready as school horses by September.
We kept some ourselves, sold most to other riding centers.
As a teenager I was the test pilot in our feral horse training, after having started a handful of gentle colts, that rarely acted up or bucked.
The riding instructor would do the initial handling, showing us how all along, wish we had videos then, wonderful teacher.
Once gentle enough on the longe line, he would give me a leg up and I was test dummy/passenger up there and followed his instructions.
Generally we had the horses, males still stallions and some maybe 8-9 years old, gentle enough to get on by the second or third day, some we took longer.
We thought we had a good system all around, the idea was, a good job is one where horses learn without fireworks, if something went wrong, horse would buck, it was our fault, we overdid it past it’s comfort or we had not prepared horse well enough.
There were a few exceptions, the one time at the first ride, horse properly prepared we thought, trainer gave me a leg up but somehow horse shied forward and I landed, on my knees, on the horse’s rump, holding onto the cantle of the saddle, on a scared horse bucking.
Instructor had the longeline short and shut him down right away, but I was launched off first buck.
We regrouped, worked with the horse again, realizing we had found a hole in our training, we had not been desensitizing the back of the horse enough, we needed to work on that longer.
Before getting all the way up, we extended our body work do more on the hind end.
There were always lessons we learned all along, the instructor always making a point everyone was learning all the time, from every thing, being observant what made a good horse person and better horse trainer.
Not sure the average student needs to be learning to handle bucking horses?
Sometimes, horses buck because it is their natural expression when happy, when stressed and when startled and when hurting, of course, always keeping that in mind.
Also as trained on purpose, as rodeo bucking horse, or trained by mistake to buck, as when saddling first time and hazing horse loose around to “buck it out”, a common practice that never made sense. It lets a horse practice a habit, bucking, we definitely don’t want in a riding horse.
I would think the OP’s trainer has evaluated the horse, maybe feels it’s bucking is something the OP can handle, or learn to handle.
Even then, the OP has to ask the trainer, maybe watch the horse under someone else to decide if handling a horse that is known to buck at times is something she wants to learn.