I'm sure, by the sound of it, your father is a good horse starter. Bet he hasn't broken a horse yet
Horrible term, "horse breaking".
Yes I use one rein stops all the time. For a young horse they make them very much less claustrophobic and give them less to push on.
A good question to ask yourself is "why would pulling on the mouth straight backwards cause the horse to stop?" It is because we have always been taught that this is the case that we assume that the horse "knows" this as well. In fact the horse has to be taught that a backward pull on the mouth means stop, it is not instinctive. Their instinct is to push into pressure, it is called "opposition reflex". So if the straight backwards pull does not work then a one rein stop is in order.
A one rein stop involves bending that neck around so that the head is not facing forwards. With the head facing backwards it is difficult for the horse to go forwards and they will tend to circle to a stop. The trick then is not to release the pressure until they are a) stopped (and I mean completely), b) they have yielded to the pressure and the neck is nice and soft in a bend, c) they are mentally thinking back to you (if you can see white in the corner of the eye they are still thinking forwards as their eye is turned that way).
This relaxation in a bend can and should be taught at the halt. Parelli call it a neutral lateral flexion. The idea is to release all tension and intention from your own body, then take hold of the rein quite near to the halter/bridle and ask the head to come around in the above mentioned way. It is not just the physical we are looking for it is the mental as well, and that is important. The horse must be thinking back to you. To start you may only get a few degrees of bend, but slowly this can be built up to the nose being nearly on your knee for up to a minute at a time.
This is an exercise I do EVERY time I get on a horse just to be sure I have brakes before I move off. With a trained horse it takes seconds, with an untrained one it takes as long as it takes.
James Roberts, my late instructor and a professional colt starter, did all of his first rides on young colts with just one rein. It was just a rope halter and a twelve foot lead rope. This was so that if the colt got upset he could not go into predator mode and try to pull on both reins to stop them, he had to use a one rein stop, it's all he had
Whenever I saw one of the colts get upset with his apprentice riding then you would hear James shouting "bend him to a stop Josh". As they got a lot of troubled horses and restarts in you heard this a fair bit when visiting them !
He suggested I do many hours of riding on our well trained horse, Bonitao, with just one rein and the halter. (In a safe environment I must add). This was to build in me the muscle memory of stopping a horse with just the one rein. I aspire to become a colt starter myself so it was especially important training for me. In fact I found it made me much more fluid in my whole riding style and I still go back to doing it from time to time. Snag with Bonitao is that he has big ears and I kept getting the rope caught on them as I changed reins. And I was expected to change reins on a weave pattern at trot
Once the horse is well trained then one rein stops should be rarely needed of course the stop should just come from your body energy and seat not the reins. The reins should only be used if the horse does not stop, initially with even pressure on both but with just one if the horse still does not listen.
If you pull back on both reins on an upset horse and get the neck flexed down then of course you are in a perfect position for a buck. If the head goes up then a rear is in order. With a one rein flexion then neither of those scenarios are so likely.
This is actually a good way to teach the horse to buck or rear. The horse gets upset and tanks of, the rider puts loads of pressure on the bit by pulling on both reins. The horse bucks/rears and unbalances the rider who momentarily releases the pressure on the reins. The horse learns, according to the principles of negative reinforcement, that the pressure goes away when they buck or rear and so "learns" that this is the correct response when galloping to pressure on the bit. It may take just one experience for this to be learnt and a long time to unlearn it. Another good reason for the one rein stop as the other hand can stabilise you in the saddle and you are less likely to become unbalanced and give those little inadvertent releases of pressure which teach the horse a wrong response.