This is where the idea that people who've only taken lessons can't really ride comes from. Beyond novice level, lesson horses shouldn't be push-button...Not giving riders experience with bad behavior is doing them a disservice.
The schools I went to would take a horse with any habit except rearing - Everything else we were expected to deal with.
Nor have I ever met a push-button horse. They may exist, but I haven't met one. No one will ever accuse my Arabian mare of being push button...panic button maybe, but not push button! Nor have I had lessons. Not from a school. Just from my mare.
Meanwhile, if I pay someone for a horse to ride, I expect one that doesn't continually bolt. If someone's horse needs basic training, and they want me to do it, then they can pay me. Until then, they are working for me and can provide an appropriate service or I'll take my business elsewhere. If the OP is continually falling off, that is NOT appropriate service.