There is nothing the OP is missing, it is well thought out and quite accurate.
Ok, lets start with why a caveson is on a bridle. Military riders used them to tie. Dressage riders initially rode in a lunge caveson, and it was used for lateral flexibility (no snaffle), and curb came for later nuance.
Fast forward (1700s), when progressive training came in a snaffle (fulmer). And a dropped was added only to prevent possible crossing of the jaw with horses being progressively trained ifv. And thus it continued until the late 60s when (german) auction horses were sold out of the country. Viewers didn’t understand drops, so the ‘cross between’ was the flash. And they became ‘in style’. Just as copying square saddle pad (sheer use was to keep tailcoat dry…now EVERYONE uses them even hunter riders LOL) or unbraided forelocks (one winning stallion who did not allow handling there).
From riding in both, there is simply no reason for one. If the horse is ridden progressively (up/open/ifv/with a mobile jaw) there is little reason for the horse to open the mouth. And if the horse starts to have mouth issues it is INFORMATION. If the horse has mouth or tongue issues it is often because the horse is steadily at (or more) vertical and the bars are exposed to the effects of the hand, rather ifv with pressures on the corners of the mouth. Horses ‘evade’ pain, and riders should ask different with their aids rather than strap the mouths shut.
What should stabilize the bit? That it is high enough (a wrinkle or two in the corner of the lips) and steady riding ‘in position’.
What does the lower jaw need supporting? A half halt should not work against it, but on the corners of the mouth. The jaw should be mobile/allow for chewing and swallowing. It should be neither clamped shut nor hanging flaccidly, errors in either are information to the rider about the texture of their aids, and the horse’s balance. Horses only clamp their jaws to evade pain. The rider should be holding both sides of the bit evenly, so it will not slide to one side or another, and in a fulmer or full check it cannot do so.
The way the rules were changed is somewhat of a joke, since the fingers now go in the side (where the bones are) which can allow the fingers even if fairly tight. The fingers should go under the jaw. But interestingly enough even with those things many horses in curbs now show open mouths or bitten/bloody tongues because now they are attempting to chew in their closed postures.