I have a younger sister that worked for a few years at a horse rescue, training their horses after they had gained enough weight. Some of which were unsound, for very light riding or could only carry a one hundred pound rider (child). Most of the rescues had huge trust issues with people, and weren't even halter broke at 6+ years of age. They did have the papers on any registered horses that they would give you after 2 years of carring for the horse. They had a registered arabian mare at one time that had come in very pregnant with a purebred foal, but even if someone adopted her and baby, they wouldn't get the papers for two years and it would cost quite a bit to register the baby at that point. And you would never be able to breed either of them EVER because of the adoption contract.
My oldest sister owns a beautiful grade arabian mare who is almost 26 years old. My mom had bought her and another grade arab mare (from the same seller) at auction for $50 a piece 24 years ago, saved from the meat man. She bred both mares twice (to a black QH stallion) and raised all the foals with her own 8 children (both fillies she got breeding to the QH stallion have passed away now). After the grade mare was given to my oldest sister as a gift, she bred her to a grey arab stallion, and got a registered half arab daughter that is as sweet as honey, so docile, calm, and gentle. My sister took her horses to college, and have always been with her. She bred the daughter (a maiden mare) the first of June as her last chance to get a foal out of her beloved horses. She is confirmed in foal (6 months along now) and will be a young 18 year old mother.
No abused/neglected/starved horse could ever replace the highly anticipated baby. My sister will pay more in total costs of raising a foal to maturity than buying a young ready to train or green broke horse off the horse market, and way more than adopting a rescued horse.