Your poor DD, sorry she's had this experience
I've said this on other threads and it isn't pony club bashing at all ( I have been involved with some very good ones both for children with and without own ponies), or having a go at you OP (how would you know after all!), but I would never recommend pony club as a child's only tuition/only riding time. Pony club is great for fun, extra riding time often at reduced cost compared to 'proper' lessons, games, team events, exams etc. but there is rarely the time for proper good quality teaching of new skills, and with lots of DC often of differing riding abilities all in one group, chaos can descend quickly as you've found! So they really need to be having at least the odd proper lesson elsewhere in order to benefit from PC.
I would do as others have said and book her in for a course of 6-8 private lessons at a really good school (I wouldn't go back to the round and round in circles ones either)- I'm afraid you are going to have to pay, perhaps as much as £40-50 for a half hour, and perhaps travel further afield if your local schools are no good. But view it as an investment in your DD's confidence and future enjoyment by getting a proper experienced teacher and a very good pony who is the right balance of calm and responsive! The aim is that in 6-8 weeks time she should be to be able to walk, trot and canter independently with a good secure seat (no holding on!) - but she might need to go right back to basics and do some sessions on the lunge or with a helper just to improve her seat and get some confidence back up. Even if she doesn't crack the canter, just walk and trot with steering and brakes installed, off the lead rein and not following a pony in front would be good and really set up her for the future.
After the 6-8 weeks the teacher can hopefully advise you on getting her back into a small group lesson with other children of a similar ability - I do think this is the best way for DC to learn if they can only ride once a week, at a well run school it shouldn't be 'follow the leader' for the whole time (or at the very least they should take turns being the leader!), they can work as individuals, do gymnastic exercises and stretches, pole work and jumping, mini gymkhana games, little hacks etc etc - much more fun than riding alone the whole time!