This question is like asking how long is a piece of string.
There is a lot of variation between riding schools and the approach they take. Some don't really believe that children below 7/8 should be doing anything remotely demanding. They will effectively babysit your child on a moving sofa.
Some children are naturally more cautious and dislike the feeling of trying to balance. They need the moving sofa approach.
Many schools find it very difficult to have well schooled tiny ponies because tiny adults are hard to find. And tiny ponies are often naughty. It's much easier for them to put a small child on something bigger, explaining away the fact that the child's legs don't reach past the saddle pad by promoting that pony as a saint. IME that pony is probably not so much saintly as dead to the leg, bored stiff and will only go up a gear after a lot of unsettling hauling at its mouth by a leader or trailing a whip behind it. Or having an older rider with a crop and strong legs on top. I don't see any children progressing well in that environment.
I think you need to work out if your child is about the equine therapy or the riding. It can be both. But if it is about the equine therapy and she is happy and the children are engaged with well, I wouldn't care about her progress just now. If she is a little frustrated and losing interest, you might need to think again.
IME, children learn exponentially faster in private lessons. They're often given a more responsive horse to ride and they do a lot more work during the lesson. However, if your child is cautious and happy in their comfort zone, I'd let them stay there and make it clear to the instructor that this is what you want. With that said, my child's additional needs were greatly helped by her advancing ability to ride around the age of 7. Following a sequence of instructions, noticing the letter names around the school, wobbling and practicing left and right were all helpful skills. I don't know the details on the wobble board research but I do know that my child's academic progress leapt forward when she was riding several times a week. Her right and left was noticeably better. She was more alert to the world.
There are schools out there with very patient bigger ponies who tend to be responsive to the instructor's voice commands which allows the child in a private lesson to feel they're doing well and actually progress rather than kicking fruitlessly. That's ideal.
We went a little overboard and bought a Dartmoor show pony who had the reputation of being a saint and he was. The are plenty of exceptions but as ponies go, Dartmoors are the golden retrievers of the equine world - placid, easy to train and fairly food orientated. We also bought a larger pony who was a witch in the stable but adored having a tiny person on her back. By that point we were spending huge amounts of money having both ponies on livery and being schooled as well as the private lessons on them - but my daughter was so happy, engaged and the riding time distracted her from the huge amounts of extra learning time she was putting in at home. She was obsessed and her ponies were her world. I wouldn't do that differently.
In terms of progress, she was flying around after six months, competent and relaxed in canter on her Dartmoor. Or as competent as a child can be when they are highly amused by the flapping of their pony's mane and don't care what direction they're going in. The bigger pony was a jumping schoolmistress and loved flatwork - figures of eight, changing diagonals, obstacle courses all at a very, very swift trot. No other pony could keep up with them.
She then had a fall that was worse than the others had been and tensed up. She continued to love schooling and hacking but rarely cantered. She didn't ride as often. That was fine until she was 13. By that time both ponies had been outgrown. The pony obsession returned with a vengeance and she was back to two private lessons a week. She then decided that she wanted to do all the things, worked very hard and was galloping happily up mountains and along beaches with her riding school within five months of returning to the sport. She is clever but naturally nervous and her bond with the pony means a great deal to her.
I hope this helps you to see that it's all about what is right for the child at that time. There is no one size fits all progress. You will feel if your child's needs are being met. With that said, my daughter did have a very strong bond with one instructor who understood her perfectly and insisted she try new things to find out she could do them. In the context of a private lesson with a motivated, sympathetic instructor, that was fine with me.