"On the bit" is a phrase I try not to use. Instead I will use "in balance", "connected", "accepting contact", "soft in the bridle", "round" and other semantic terms because, truly, the bit has very little to do with it. In all actuality a horse can be "on the bit" in a bitless bridle, or a halter, or nothing. It's a feeling, more than anything, that the horse and rider are of one mind and one body. There is no resistance or tension from either participant and by merely thinking of what to do and where to go the rider gets immediate cooperation from the horse.
Just because a horse puts his head down or arches his neck does not put him "on the bit". A lot of false postures can happen through strong bits or ropes and pulleys or restrictive gear that can cause a horse to adopt an appearance of being connected but in those cases the horse's body is not involved so he will still be bracing and uncooperative.
Other phrases I don't use are "head set" and "in a frame". Both sound very fixed and rigid to me. It may be a case of tow-may-toe, tah-mah-toe but I just can't bring myself to say the words when referring to a horse's balance.
A student once asked me if a horse ever learns to just stay on the bit by itself. I told her no, that a horse can only respond to particular cues when applied. That's not entirely true. Oops. Should have thought my reply through a little before answering. When left to its own devices, a horse, on its own, of its own free will, will move on the bit. Horses arch their necks, collect their hindquarters, lift their backs and move across the ground as if on air when they are feeling particularly frisky or showing off for another horse (and sometimes just showing off in general even if no one is looking) and that is what we strive for as humans: to ride a horse in a way that looks as if the horse is thoroughly enjoying himself and having the time of his life.
How is it possible to achieve that through a strong bit or restrictive gear? Does the horse say "My head is tied down, and I am having the best fun!"
I'm not against bits completely. I do wish certain bits were illegal to use in competition and that more bitless bridles were allowed but I do ride most of the time with a bitted bridle and I try to do it as compassionately as I can. I expect the same from my horses as I give them which means there is no yanking on the reins. Unfortunately, some horses, either through lack of training or poor training somewhere in their past, have learned to yank or lean on the reins. Those horses, I call spoiled, not spoiled like a bratty kid but spoiled like a tomato left out on the counter too long.
With work, those horses can be reclaimed but on the condition that any riders from that point on, ride them with skill and don't let them slip back into the bit-pulling habits. Horses don't learn to lean on the bit by themselves. It comes from a rider who is heavy on the reins or a horse that keeps the reins heavy as a defense against rough hands.
In order to re-educate that type of horses it can be necessary to take the bit out of the equation and ride in a bitless bridle. Other times, it means a change of bit. But sometimes the habit is such that regardless of what is in or not in the horse's mouth, he will lean on the reins. A human's hands can never be utterly steady and consistent so it can be very difficult for a person to work through that issue. A rider should always strive to work on having steady, controlled hands - which doesn't mean fixed hands. Fixed hands are dead hands, they just plant and stick in one place. Fixed hands can be a step toward learning to keep hands steady and is a good lesson for a lot of beginners but at some point, the hands then need to come to life and communicate with the horse. One thing at a time.
For riders who haven't learned that nuance yet, or for horses that are particularly strong, I will use sidereins on the horse when it is being longed. Side reins are not for riding in, that is a recipe for disaster. Sidereins used when longing need to definitely be used with care. They have to be introduced gradually, and fastened in such a way that the horse is not forced into a position but when he pulls on the reins aggressively, that the reins are tight and uncomfortable. As soon as the horse relaxes, the pressure on the bit should come off immediately. Just as with any piece of equipment, sidereins can be used wrongly and cause the horse more grief.
There are always going to be spoiled horses (which keeps horse trainers in business) just as, at least in this house, there will always be spoiled tomatoes. At least the training I do allows me to be able to buy new tomatoes. Buying a new horse is not as easy.
However, a spoiled horse can be reconditioned, unlike a spoiled tomato so there is not need to get a new horse. Don't think of your horse as disposable, but put some effort into becoming a better rider, trying different or less equipment and get your horse out of a frame and into a conversation. With some time and consideration you and your horse will start speaking the same language and you will both be having the best fun.