We always ride with a halter underneath.....well...not always...but usually. I am always tempted to take it off but if that bridle breaks or I pull it off when falling... I want to be able to catch the horse.
At the chuckwagon races I saw 2 horses running around without a rider or bridle.
The first one came bursting into our camp all wacked out and scared and I had no way to catch it! It just ran back out of camp and the fella got it later. And the 2nd one I saw was involved in a horse race. The horse stumbled and fell at a full gallop and the reins got tangled up with the rider and the horse flipped over (it was aweful) The horse got back up and continued to gallop off but no one had anyway to catch it. (the guy was ok too)
So for safety I do it and also because I don't like leading the horses with a bit. I'd rather use a halter. I think its nicer.