While I agree with the sentiment, I don’t think riding the small selection of beginners ponies found in your typical UK riding school in weekly lessons where you are never allowed to do much anyway because ‘health and safety’ is ever going to produce a really competent rider.
Getting a pony doesn’t have to mean the end of variation - on the contrary it can open up more opportunities for this. Our pony club rallies always feature a bit pony swapping, plus meeting fellow pony owners will inevitably result in more casual swaps and ride offers. Child could end up riding anything from a top class schoolmaster to a newly backed youngster.
11 is a perfect age for a pony, I was around that age when I got my first one and it was the making of me. Horses ended up my career as well as my passion. Just make sure that said pony is the right one. So so easy to make an expensive mistake, especially if you lack experience. Lots of unscrupulous sellers out there that will say anything you want to hear - never believe anything you are told, always verify for yourself. Vet check is important but that only covers health and soundness. You also need to make sure that the pony is good and safe to ride and handle in all ways - hacking alone and in company, open spaces, loading… anything that you might want to be doing.
Admittedly the approach to winter isn’t a brilliant time to buy a first pony, although the livery yard plan could work well here, assuming the yard has a good floodlit ménage (or even better, an indoor arena!)
Finally, be sure to factor in transportation and pony club fees into your calculations. It gets lonely and boring very quickly just riding at home and having nothing to aim for. And obviously she’ll need to keep up with regular lessons too. Good luck!