My dd is nearly 8 and has been riding around 20 months and this is our experience.
Started at 5, when a friend used to put her on the lunge initially and we did this, intermittently, but she would also take her on walks (on the lead rein around the fields).
She then started at a local riding school in a group lesson, each had a leader and did this for just over 6 months. They did walk/trot, but no canter and she was getting bored. I suppose this was for around the same time (once a week x 6 months) as your dd. Progress was really slow and like your dd, she was desperate to be off the lead rein, but this probaby only happened two or three times and then the leader would walk beside them, but the ponies wouldn't trot without the leaders and dd ended up doing a lot of -booting-kicking, which I was not happy about, so pulled her out.
At the beginning of the summer holidays we loaned a horse from the yard next door to us a lovely sweet pony and I have been teaching dd until she has had a a couple of private lessons recently.
I kept her on the lead rein for a few days when we first got the horse, (still do if out for a walk with her), but literally within a week she was off on her own around the school. She was so obviously ready, and quickly trotted confidently. However, we have spent a lot of time working on the basics and it is only recently (nearly 3 months after having the pony), that she has cantered which she did on her own decision.
I would say your dd, especially as she is older than mine, is certainly ready to be a bit more independent with her riding and definately look for somewhere else.
There is a lot to be said for working on the lunge, we still do a bit - when I can persuade dd - as she wants to do 'proper' riding as she calls it!!!
We are in the northwest, not many good schools around this way either.