The SPanish train their Andalucian stallions with a Cerreta - a fiersome apparatus which often leaves a scar on the nose of the horse. Some trainers, when basic training is completed, follow it up with a hackamore of various patterns. But in Spain a lot of men ride horses invariably stallions which you'll see happily mixing with each other at the ferias (fairs), A Spanish trained horse is obedient and is very often a horse which is a delight to be ridden for any purpose by a skilled rider.
Monty Robert's "Dually" halter works by pressure on the nose,
Personally I have a rope training halter, bought years ago in some Western tack store, which works by pressure on the nose and the poll. I use it especially for training work on the ground.
On a well schooled horse, the bit should be seen as a communication aid, not a set of brakes. So if you take the bit away, then you lose a degree of communication.
However if you are going to feed the horse up with high energy rations and then take it out on a hack with a group of horses for a burn up, then the horse will get steamed up. And then the rider might be gratefull for a more powerful stopping bit or perhaps at least the leverage from a running martingale.
Bits and bitting is a very specialist subject but many horse behavioural problems are caused by incorrect bitting and
poor hands. It is worth getting to understand some of the principles.
There a good website www.sustainabledressage.net
which discusses bitting. And the Myler Bit company explains their ideas on their web site.