I don't think much more needs to be said about the weighing - I'd only add that I'm surprised the riding school will guarantee/ promise her to ride a particular pony even if she isn't too heavy. It's the biggest nightmare matching horses to riders (not just appropriate weight/height but workload, abilities and confidence too), at every school I've been involved in there's a constantly moving and delicate equation of which horses are a bit lame or off colour this week, which are too fresh or need remedial schooling and can't have beginners or nervous people on, which riders need a step up or challenge and which need something super quiet/safe, that there's simply no room for promising kids they can definitely have a particular pony for pony week, even if there's no issue whatsoever with them fitting on the pony physically. The schools I know apply the 'no requests, no preference' rule pretty strictly because even if the child's favourite pony happens to be a less in-demand one you don't want to be (seen to be) treating one child (or parent!) differently to another or allowing wriggle/negotiation room. But hey ho, everywhere is different I guess.
And as for the 'schools should just have horses to fit every size' argument, I just don't think it works in the modern era. Even assuming you can fairly accurately tell someone's weight by eyeballing (which I personally can't), I just don't think riding schools can feasibly keep enough of a stock of big horses to fit the number of big(ger) people who want to ride these days. I think it would be fair as a finger in the air estimate to say that many women who would like to take up riding as a beginner weigh in the region of 13stone (say 80kg). That's not someone who would be obviously obese to the naked eye, they would probably look quite average and wouldn't be very far into the 'overweight' BMI calculator if they're quite tall. Add 12kg for clothing, body protector, hat and tack (conservative estimate) and you need a 625-650kg horse - and I would say that's really a bare, bare minimum, the 15% 'rule' is a guide and a maximum not a vague indication, a lot of people say it should be closer to 20% especially for beginner riders who may be more unbalanced and where carrying a heavier rider is the majority of the horse's work. So you are looking at a 16.2hh+ draft type horse - horses of that type suitable for a beginner rider are very, very expensive to buy and keep, to the point where most people absolutely couldn't afford what lessons would cost if it reflected the price of that horse. Of course some schools do have these types on their 'roster', many would have at least one good solid weight carrier but the trouble is (a) if the one horse you have for that client group goes lame or becomes unavailable or simply unsuitable for that rider for other reasons you have to let clients down (b) as the horse will have cost you so much at the outset and are hard to replace you're inclined to be particularly careful to look after them well and (c) if you don't have a weight limit or even have the option to weigh people and start letting anyone ride including people who actually weigh more like 14-15 stone (again hardly exceptional these days), then you have more and more clients for fewer and fewer suitable horses. It just makes so much more commercial sense to keep a stable full of easy keeping 14.2 - 15.2 cobs and natives and cater mainly for children/teens and small/ light weight adults. Sucks for the heavier people, I know, I've been that heavier person, but welfare considerations aside it's just commercial reality...