Horses are a money pit (I know, I've always had them) and for $500 you are not going to be able to get a decent horse anyway, and for that price could end up with something which is unsound and may be unrideable and cost a fortune at the vets to no avail. Unless you have your own land and stables, expect to pay up to $100 per week on stabling and food (if you look after it yourself, and many more $ if you pay someone else to do it), allocate a minimum of 1 1/2 hours per day of your own time, juggling homework, need to be prepared to sacrifice all out of school activities on an evening and weekend, and then there's shoeing at $100 every 5 weeks, horse clippers, rugs, transport, vet bills............
From your description, the reasons you give are exactly the reasons NOT to get one! And as for choosing a horse by colour - who gives a ?? what colour a good horse is? Ever heard a tipster giving a racehorse low odds because of its colour?
In short, horses are not pets but a LIFESTYLE choice and the sooner people stop thinking of them as pets the better.
I know a friend who is a knackerman and they are constantly getting people bringing horses and ponies to be destroyed just because they have bought them without realising how expensive and time consuming they can be and children have lost interest and they become disposable like discarded toys.
What your foster sibling needs to do is find someone with a horse and offer to share duties in exchange for riding, therefore they are not committed to owning one.