As loosie and others have pointed out, its rather absurd to chase a horse away in order to get it to come towards you and stay with you. In learning theory terms what you are doing is punishing it for the very behaviour you will reinforce (reward) moments later.
As loosie also points out, in the wild (or even in our paddocks) horses don't chase each other around and around. They chase each other enough to get the other horse away and then they quit.
Lastly, there is no evidence in any of the peer reviewed studies of wild horses that head lowering, licking and chewing and ear orientation are signs of a horse "submitting to, respecting" or any other kind of acknowledgement of leadership of another horse. The only reliable sign that one horse acknowledges that another horse is more dominant than it is by moving away. Not coming towards or coming closer. Avoidance is the key indicator horse researchers use to tell who is top horse and who is not when studying wild horse herds.
Many species of animals will lick and chew after they have received a fright, including dogs, cats, cows, goats, sheep and horses. Its thought to be due to the drying effects of adrenaline on saliva production.
Lastly, Join-up works for the same reason that any horse training system works- negative refinforcement and classical conditioning. There is no mystery or magic to its effectiveness and it has nothing to do with the horse thinking that the trainer is another horse or that the trainer speaks its language. We lack a lot of the features that horses use to communicate with each other, like moveable ears, heads on long necks, tails, four legs etc.
If you really want to bond with a horse using food rewards to train it to perform specific behaviours is ideal. You will be giving it something it genuinely likes- food and it will associate you with getting that food. Getting chased is not rewarding for a horse and given we need our horses to stay with us rather than run away it is an illogical way to teach them stay.