Every bit of a horse's coat, mane and tail serve some protective function. But stalls, blankets and shelters serve to protect them, too. If you cut the mane bc of burrs, or some other reason, it's just a bad haircut, and will grow back. =D
BTW, Native Americans put fringe on their garments (which Mountain Men, firstly copied) to mimic wild animals. Rain sheds away from the animal's coat, and sheds away from the garments and the wearer and that's how the mane helps the horse, too.