Your daughter sounds like she is a real horse girl, I love it!
From the videos, I can very much see how she would have been pulled out of the saddle by the pony dropping its head. She does have the typical beginner balance, hasn’t figured out how to use her core effectively and have her weight in her heels. Things she can do now to build some muscle up while being unable to ride are squats and wall sits, calf raises on stairs (really focusing on having the balls of her feet on the edge of the step and letting her weight sink into her heels so she starts seeing what that should feel like), she should be able to do these with an injured wrist and it not be too “boring”. She can also do balance work too, like tree poses, practicing standing on one foot and looking straight up, sitting on one of those exercise balls and picking her feet up to try to balance on top of the ball. All of these will help her find her core.
I agree that the barn itself doesn’t look awful (I know several in this area that are so much worse that a lot of parents think is amazing), but if your gut says its not a good fit, listen to it. I do have to say though, if the trainer that was teaching the lesson is new, there is a possibility that she might not have as much experience teaching beginner riders and wasn’t sure how to address the problem with the pony beyond getting back on and trying it again. If she is a natural rider, it’s hard sometimes to remember that not all kids have that stickability factor naturally, and she might not know how to teach that. Giving lessons has a big learning curve.
If I were you, I would contact the barns in the area and talk to the instructors. Let them know that your daughter absolutely loves to ride but is having some issues with her basic position, strength and balance that you would like addressed before moving on to anything more advanced. Ask if you can observe some lessons, and choose the barn with the relaxed, quiet horses and ponies, happy looking kids, lessons that will do a lot of two point and no stirrups work at a walk and trot, little games like “around the world” that help teach balance, things that address the riders’ positions and not just trotting and cantering laps around the edge of the area and over poles saying “good” over and over. You should hear things like “Sally, don’t look down, keep your eyes up!”, “Heels down”, “Shoulders back”, “Use the neck for balance not the reins”, etc quite a lot since kids need a ton of reminders before muscle memory kicks in. Some of it will just take experience as well, learning what a good instructor is like. But in general, a good barn will address the weaknesses of your daughter’s riding base first and foremost.
And if you are ever not sure, you can always come here and ask for advice. There are SO MANY experienced people here, and while there are always some that are inexperienced that try to chime in, in general you will get a good consensus from the majority of people.