I will agree with your trainer on one thing: it’s not the pony’s fault.
It’s the TRAINER’S fault.
Now that we have the full story (at a walk/trot, and fell off over his head), the entire thing is the trainer’s fault. Not your daughter’s. Not the pony’s (who could either have just been acting up that day as we all know ponies do, or may be tired/sore/etc., which again, is something the trainer should have caught).
Your daughter needs better instruction, on a regular basis, with someone who specializes in starting green riders correctly. Was she ever started off on the lunge line, without stirrups. to help her develop her seat and balance? Does she know how to adjust her stirrups properly? Does the trainer encourage them to ask questions in lessons?
She might be one of those naturally good riders we all envy :), but she’s still inexperienced. A better instructor will be able to take that raw talent and turn it into something great.