No offense intended, but I think it's a little too much of an accusation when you said the riders hang onto the horse's mouth. It's not intentional, if you stumble on the stairs the first reaction is to grab onto the railing. Same thing when you ride, that's the reason beginners have a hard time without the reins, they use them as balance.
I actually always blackout when I fall, the minute I know I'm getting off, my system shuts down and I wake up once I'm on the ground. I might scream, grab the reins, or do something totally stupid, but those occur when I'm blacked out and therefore if I grab the reins, it's just a reaction that I can't control because I'm not choosing to shut down, my body is.