Just as your son may be a good runner, A track coach can make him into a great runner. Gaited horses do gait naturally, but we as riders can coach the horse to improve. We can help the horse to hold the gait we desire to ride.
Do you even know what is is to "pivot"? If the bit does not move but the horses head does then the bit is not a point of pivot, it is the point of leverage and the poll is the point of pivot.
Yes I know what a pivot point is. Do you know what a head nod is? When gaited horses gait, they use their head to help balance their gait. A trotting horse does not nod his head. Since the trot acts to counterbalance itself. In fact many judges or vets look for a head nod as a sign of lameness in a trotting horse. A head nod is where the horse uses his neck and head to counter balance the moton of his hind legs. It's more of a hammer movement of the neck. Gaited horses are typically ridden with their nose at or behind the verticle. If they were pivoting at the poll, there nose would be in front of and behind verticle. We refer to breaking at the poll as a verticle flex, not a nod. With the horse nodding while gaiting, and with his face staying at or behind verticle, the bit becomes the pivot point. This is way gaited riders maintain contact with the horses mouth. They typically don't ride with loose reins. But rather maintain light contact. The rider holds steady hands. They are not moving forward and back in time with the head nod. The hands should be held still. So if the hands are not moving and you have tight reins with mounth contact, and the horses head is Nodding, The pivot point becomes the bit.
So which is it? Collected or inverted?
This can be done in a snaffle, but it is just easier to use a leverage bit to force the horses head up and the spine to invert bringing the front end up to better "enhance" the gait
The horse is collected on the bit to get the horse to come to verticle or slightly behind the verticle. This raising of the head and tucking the nose behind verticle causes the back to invert which you descrbe as bringing the front end up and enhancing the gait.
There is an old TWH saying "That if he's not shaking, he's not walking" But not all gaited horses nod or shake their head. Paso don't nod, Rocky Mountain Horses don't nod. Their gaits don't require the head nod to counterbalance the leg motions. Many foxtrotter and TWH don't nod or have very small nods. So it's not manditory but most judges like to see it.