Horses are tougher than they look. Millions used to live in awful conditions before the internal combustion engine filled the world's glue factories. Now, in the absence of wolves and mountain lions they're overflowing their bounded western world and the private lands that stockpile the overflow.
Modern Americans are more sentimental than they were 100 years ago, so we're unwilling to shoot them all. Were I a horse I'd rather be shot than starve or be eaten alive by wolves, but nobody asks the horses.
So why can't we hire cowboys to shoot mares with Depo-Provera in late May? It's cheap stuff, its used with horses, and it's designed for deposition. Shoot a capsule of it into the mare buttocks around mating time.
It's been done for lions.