Absolutely it's fair to encourage her, lots and lots of kids never own their own ponies and still get plenty of enjoyment and benefits from riding as a hobby. Some will go on to be horsey adults and buy their own horse or work as professional in the industry and I always think it's really good motivation to not have had it handed to you on a plate as a child but to have had to work.
Please 's advice above is excellent, if she gets really into it so a weekly lesson isn't enough, look for a large riding school which does 'own a pony' days or 'pony weeks' in the school holidays or which offers Pony Club - this is like Scouts or Brownies but with ponies, they do ridden activities and games but are also taught about pony care and stable management, there are 'badges' to be earnt and also often an annual camp. There are branches for children that have their own ponies but also ones for those without, the latter usually attached to larger riding schools so lookout for that when choosing a school.
When she gets a bit older, as Please says, a lot of horsey teens particularly those without their own horse, earn extra riding/horse time by helping out at their riding school doing chores and yard work and helping the younger riders, usually they aren't paid in cash for it but can be given free or discount lessons or just get extra riding in by exercising the newer or naughtier ponies
. This process has become a bit of a rite of passage for teens in the horse world, it's so good for them, they learn about the harder more boring side of keeping a horse (endless mucking out, sweeping the yard etc), the value of commitment (the ones that reliably turn up in all weathers and don't complain etc are usually the ones given preference for 'extra' opportunities!), gain some leadership/teaching skills from helping the littler ones, make lots of friends and have a lark etc.
Kit wise you've had some good advice, the only thing I'd buy new is her own hat (once you are sure she's sticking with it), take her to a proper tack shop to get fitted as it's so important for safety and comfort. To be honest the fact the school let her ride in a hat that was too loose or too tight worries me a bit, who fitted her for it? The school should have a range of sizes of hat to hire so everyone should have one which is safe and comfortable (maybe not the exact perfect shape, some hats can pinch a little if you have a odd shaped head
), so a bit of discomfort isn't a major worry but letting her ride in a loose one or one so tight it gave her headaches is! Is her instructor a properly qualified instructor as if so they def should be checking that their students are wearing all the proper kit esp a properly fitting hat before they get anywhere near getting on the horse, that's like instructor 101? If the school doesn't take safety seriously I'd be looking to take her somewhere else IMO - a properly regulated BHS or ABRS approved school is what you want...
Otherwise everything can be bought 2nd hand pretty much and there are thriving marketplaces on ebay and facebook marketplace for kids riding gear. Check with her teacher/the school about their 'rules' on body protectors and crops, the school I am based at they don't 'have' to wear body protectors except for jumping and hacking (which will be a way down the line for your DD) so that may or may not be a priority for you, and beginners don't carry crops so again you may want to hold off there. Some second hand kids jodhpur boots, maybe suede chaps, gloves (don't have to be fancy riding ones, cheap woolly 'magic gloves' are just as good) and some jods or riding tights will almost certainly be a good investment though - riding as a hobby is a fruitful source of presents as you'll find, there's a whole world of stuff you can get her!
Sorry for long post, do come back with any further questions, this a friendly and helpful board!