First, please take this as me being completely honest, and not trying to be mean in any way. I know it can be hard to hear someone critique your child.
I haven't watched the entire first video, but from the first few minutes, I personally would not let your daughter on a young horse for these reasons:
She has very unsteady hands. See how she keeps lifting her hands everywhere to adjust her reins? That is not helpful in training a young horse. A young horse needs someone who has quiet, steady hands, who isn't accidentally jerking and pulling on their mouth. She also has a very unsteady seat, which can lead to a frustrated pony, because she can not give steady and correct cues as long as her seat is all over the place.
In posting the trot, your daughter is leaning on her saddle for balance. This puts awful pressure on the pony's shoulders. When she takes her hands off the saddle, every time she posts she catches the pony's mouth with the bit.
For a young rider your daughter is definitely NOT a bad rider, she is actually quite good for her age. However, children at that age do not tend to have the necessarily motor skills to ride young horses, and I personally would find her a very well trained, push button, school master. I do not think she needs a dead-head, idiot trained horse at all. But I think before she steps up to a young, green horse she needs to ride a push-button, meaning a horse that is still sensitive enough to give her a challenge, meaning she would have to get her cues and seat right to achieve good results with the horse, but something that will also be forgiving of mistakes.
While this pony may be good-natured, eventually, it is not going to be too fond of someone bouncing around and jerking on her mouth, and she will react to it.
Perhaps finding a nice push button for your daughter to practice her own equitation on for a while would give her the skills she needs to bring a young horse around.