Frankly I think she rides better than some adults.
There are some obvious things wrong with her equitation, but I don't think there is anything that is going to ruin the pony.
I'm assuming the narrator is also working with the pony and it is not just the child training the horse (I could be wrong). Hopefully this is the case as there are some things that a child of her capabilities will not be able to teach or bring out in the horse.
Ideally an adult would be the primary trainer, but letting this girl help right now seems to have worked so far.
Sure it's a huge risk to put a young horse and young rider together, but I think if you are going to do it, this is the best example I've seen of it in awhile.
I do want to add just to make sure you're not treating the pony as a broke pony. I've ridden some that will trot around like that on a loose rein, and then when they start to learn and gain experience they just turn into monstrous things for a ride or two to see what they get away with. I'll admit I was a nervous wreck when she was desensitizing the pony to the jacket and had her hand on the buckle. If the horse would have decided something was going to eat it and taken off, I'd be concerned about how she would recover.