I think you are being very sensible to be cautious about buying, a horse or pony of your own is a huge practical and financial commitment. Your DD will have GCSEs around the corner and bear in mind that not all pony-mad kids turn into pony-mad teenagers, there are so many distractions for them at that age!
I think a loan or share would be a better first step, you have a few options to consider:
-Full Loan. In this scenario you take full responsibility for all care and upkeep of the pony for the duration of the loan. You can normally decide where it is kept (some contracts will specify a certain yard). Normally there is no charge for the loan itself but obviously you need to budget for livery costs, feed, vet bills, insurance etc.
-Part loan or share. In this option the pony remains at its current yard and the owner retains overall control and responsibility/decision making for it. Your DD would get to ride the pony 1-3 days a week in exchange for a financial contribution and/or stable duties on 'her' days.
-Riding school loan schemes. Many larger schools do these now- not a true loan as the pony is still used in the school but the loaner can ride 'their' pony outside lessons for a set number of days a week. The school will often offer the chance to do Pony Club or competitions to loaners as well. A good way to ease into the world of loaning/ownership!
Some things to consider:
-Ensure that you have a written contract with the pony's owner, whatever arrangement you go for. Setting out things like who is responsible for what financially, what happens if the pony is unwell or injured, what insurances (personal, for the pony and third party) need to be in place, notice required if either party wants to end the arrangement. If you or your DD are a BHS member they have an advice line and template contracts you can access, very useful.
-Be wary of over-horsing your DD, particularly if you go for a full loan or share where the owner is not very involved/present. She sounds like a great, confident rider but it can be quite a transition going from riding well-trained, hard-working school ponies under the eye of an instructor, to riding a privately owned pony who may only be worked a few times a week all alone. There are some unscrupulous individuals out there who could try and 'sell' you an unsuitable animal and it would be a real shame to knock her confidence... for her first loan/share you should look for an older school-master type, used to being ridden by novice children (your DD may not be a novice in riding school terms but you should run a mile from anything described as 'not a novice ride' - horsey code for nutter!). Much better to play it safe and go for a steady/tolerant type, she can always trade up for something with more whizz further down the line (which is the beauty of loaning/sharing, if the pony is outgrown or not quite right then much easier to move on!). As you are not experienced yourself can you ask DD's instructor to come with you to view any possible loans/shares?
Re the age for caring, I wouldn't expect any pre-teen to take on full care/responsibility. A sensible 13-15 year old may be able to do the majority of jobs with an adult present but not actively supervising (they shouldn't be completely alone on the yard in case of accidents or emergencies). But as above, teenagers often have a lot on with school, other sports/hobbies and friends, and particularly in the winter it's hard going for them (think trekking across dark fields after school to give hay, breaking ice in water troughs at 6am, carting heavy barrows of muck and water buckets around the yard etc). So I think you probably need to count on having to provide at least some care for the pony yourself for the foreseeable future. This is why a share can be a better option than full loan (as you are only responsible a few days a week, much easier to manage). Or otherwise if you can afford it, budget to have the pony on full livery, so all jobs are taken care of by the yard and your DD would just have to turn up, groom and ride. Bear in mind also that different yards will have different rules on supervision of children on yards and at what age they can ride/be on the yard without a parent present (between 12-16 would be normal), so you need to factor this in if you were hoping to be able to drop her off and not hang around!
Good luck, hope it works out for you! Any other questions, happy to help!