Thank you! Hopefully she stays happy 
Iād say there were points not captured in that first video where she was doing it non-stop especially when I first got on and then as I asked her to slow down (i.e. half halts). Yes, the Micklem piece is quite interesting, because the seller was pretty particular about the bridle she was going in. I rode and still ride her in a $25 Silver Fox bridle from ebay, since I didnāt want to put my fancy bridle on the line with a horse I didnāt know 
I did order a used Horze dressage cavesson which marginally improved things, but by the time it had arrived we were 80% there anywaysā¦
The bit piece is actually quite important. She was in a happy mouth at the time of that first ride, and since I for no particular reason really dislike plastic bits, I started off riding her with my KK Ultra (horses seem to love it), and while it was a marginal improvement it was also too large and thin for her. I ordered another KK Ultra but now both of them sit gathering dust, because the said clinician suggested something non-loose ring so I gave a hail Mary to the simplest $20 eggbutt. D rings are also non-loose, so perhaps there you might have an answer as to why she likes it?
Are you riding western? I am probably not an expert in that case. Generally in English (or, better said, in the way I was taught) youād ideally aim to maintain a contact at all times however feather light and even with a long but not loose rein, so technically your training should condition the horse to reach into the bridle/bit at all times, even on a free walk. Now we obviously donāt always follow it, but the idea is that you always feel the horseās mouth. From a less philosophical standpoint my answer would probably be that you should teach a horse to go in a steady contact before you decide you are ready to ride without that aid. My mare is very sensitive, at this point I can ride her with 2 fingers and frankly donāt use my rein all that much, but when I do she is no longer frustrated and I think thatās ultimately the goal - to have the aid at your disposal when you need it.
Having said all of that, what confused me about my mareās case was that in fact, it wasnāt any pain that caused the head tossing, but the frustration about stuff randomly bobbing in her mouth and not knowing what sheās supposed to do and when. Nobody wants to hurt their horse, so of course having inherited the issue I tried to ride her with a loose rein and apply pressure only when needed through half halts. Most at this stage donāt get the subtler aids of seat and frankly if the horse is not balanced/on the bit itās very hard to ride just off of seat - at least for me, so I do ultimately end up using the full half halt to guide the pace. From a horseās standpoint half halts without the contact are probably super random - thereās no real way an untrained horse can see it coming - they feel nothing at some point and quite a pull the next. I mean, if someone did that to me Iād get neurotic too, especially if I just wanted to go faster and or balance my unfit butt Even if you just try to keep a still hand on a light contact, they probably anticipate pain of a sharper intervention which is why the hand needs to follow so that there is even pressure on both hands at all times. I.e. it doesnāt matter if she is bobbing, pulling or whatever - the key is to have her believe that a continuous, light and steady pressure in her mouth is normal and in fact comforting. And that is just a matter of consistencyā¦
