I was in exactly your position a few years ago. Sent daughter to pony camp for week when she was 7 and she came back very keen on ponies and has not wavered ever since, aged nearly 14.
She had riding lessons for a bit and then thanks to helpful readers on mumsnet we had 'loan' of a small 11 hands pony when she was about 9. This was the 'test' for her, I said if she could stick it out we could consider moving forward. She visited the pony almost every weekend for about a year and looked after it and rode it - we paid £10 a week to the owner and paid £40 to farrier every six weeks.
then we did a loan of a pony but it was very expensive and not great, as we could only travel on weekends to the farm where it lived as the owners weren't keen on the local livery. They wanted £400 a month.
I ended up thinking this was crazy and we might as well buy our own money for the money I was paying for her just to ride 2 days a week.
So we bought a pony at the end of 2016 for £7K and its been a very mixed experience as poor thing went lame a year later and she hasn't been able to ride him since about December last year. Insurance is essential, that has paid for his treatment.
But livery costs are only £110 per month for DIY livery and we can pay people to help out in the week - about £7 a day to take pony out and bring him in etc.
Like any such hobby it can end up really, really expensive depending on what you want to do with the horse. I was 'persuaded' to take out a loan for a horse box which did nearly financially cripple me and with hindsight was a very stupid thing to do. However, trailers are much cheaper if you have the right kind of car to pull them.
I think the key is, that given she is so young still that you don't dive into any expensive decisions, such as actually buying a pony unless and until she has proved that this is really what she is interested in. They can't fake enthusiasm about mucking out a pony in February that rewards them by biting them quite savagely!