Having worked in dog, cat, and horse rescue for over a decade, I agree with a lot of points on this thread. While I never discourage rescue, there is a LOT more background work to be done before you consider adopting an animal rather than purchasing an animal, especially with a horse.
In our state we have four state inspected, approved, and valid equine rescues. They are also 501(c) non-profits. I went through a phase a while back thinking about a rescue horse too and scoured through all four of their websites and of the four rescues in the state, I would only feel comfortable adopting from three of them. The reason I dismissed the other one was not because of the care of the horses but because of its application and terms of adoption. This facility required that the horses only be trained using Parelli (no other training styles, not even other NH methods), the groups were allowed to arrive unannounced at your facility and inspect your adopted animal, premise, etc. even if you aren't there (which I think is a huge invasion of privacy and maybe even against some trespassing laws?), and several other invasive adoption clauses. And it only listed vague, general definitions for things that could result in the removal of the animal which to me is also a red flag as basically it seemed like they could take back the animal at any time if they felt like it.
The other rescues, though, work very closely with local animal control, livestock inspectors, and the reservations here. None of them go to any auctions of any sort, and very few actually take in horses from private owners. They almost exclusively take in horses that come from legal cases or estray cases. And their adoption applications are much more reasonable; safe fencing, no riding if they aren't sound, if you have to put the horse down please let the rescue know and give them a copy of the vet's certificate, no moving facilities without letting the rescue know beforehand, they give 72hr notice if they need to inspect your facility/animal, and obviously no selling of the horse. All of those are reasonable (to me, at least).
I have more experience with dog rescue and I can tell you THAT is crazy. When I was an ACO I worked a case where a lady was literally stealing dogs from people's backyards and then adopting them out as "rescues". I've also seen breeders who list their puppies as "rescues" in order to avoid licensing as a breeder and paying fees. And I've seen "legit" rescues where applications stipulate that you must only feed your dog a certain brand of food, must take them to doggy daycare daily, must not allow them to play with certain sizes of dogs, etc. And don't get me started on adoption fees...I'm fine paying a breed rescue a few hundred dollars, but at one rescue they wanted $1800 for a purebred English Bulldog that needed another $5000 in medical work and they could not afford the medical work so instead of lowering his adoption fee that poor dog sat around miserable for who knows how long. THAT is the definition of a bad rescue.
Just do your research. Also, while the US doesn't manage animal non profits quite as much as other places, there is some federal (or maybe it's by state) website (I can't recall the name) that DOES list the finances (or at least salaries and basic ingoing/outgoing) of every 501(c) non profit, not for profit, etc. I know because at one of the former private shelters I worked at I investigated the finances because something didn't seem right and I found out the President on the BoD was making over $110,000/yr...at a non-profit animal shelter...where we only rely on donations, adoption fees, and grants to pay staff and take care of the animals. And yet the President said we couldn't afford an isolation bay for the sick animals and instead we should up our euthanasia rate...jeez. Again, just do your research if you are still serious about getting a rescue.