Howdy and welcome to the forum, Newpony
I would guess that the pony has a habit of puffing up whenever the cinch is tightened, lots of horses have this issue. Basically the only way to manage that is to check your cinch often and tighten it whenever it's needed. Saddle her up, tighten the cinch, walk her around and/or lunge her for a couple of minutes, tighten the cinch, get on and ride for a few minutes at a walk, check the cinch, etc, etc, etc.
I wouldn't worry so much about the fall so long as your daughter isn't hurt. It happens to everyone who rides
. The best thing that you can do is encourage your daughter to get right back on and try to figure out why
the fall happened. In this case, the cinch was too loose and your daughter doesn't have enough balance yet to stay in the middle without the saddle to keep her there, not to mention that the horse started going too fast. All of those things are easily remedied and now you know what to keep an eye on in the future to lower the risk of it happening again.
Personally, I see no problem with the trainer wanting her to ride a lot this early in the game so long as she is covering the basics of handling, tacking, grooming, and caring for the horse as well. The best way to get better at riding is to do a whole lot of it
I will ask though, does your daughter have a helmet? If not, that might be a good investment for your peace of mind if nothing else.
I can't say what the trainer should and should not do, but I think if it was me, I would focus on slow work for the time being. Lots and lots and lots of walking, making sure that your daughter is sitting properly and learning how to steer and stop the horse with the reins, then moving up to the trot and teaching her how to keep her body in fluid motion with the horse, making sure that she can still keep control of the pony even with the bumpier gaits. I would probably do all this on the lunge line until I was very confident that she could handle controlling the horse all on her own.