...If you are out riding and a horse spooks or bolts, will a bit really stop it any better than a bitless of equal training? Please people,dont take this question to have any hidden agenda's...
I've only used a few varieties of sidepull halters. Nothing with a cross-under design or a bosal.
For Mia, the answer is definitely yes, a bit does better.
1 - It does a better job of preventing a bolt. When Mia gets jumpy, I can play with the reins and usually calm her down, at least down to "no bolt" territory. It usually works well to put a pinkie of pressure on each rein as that shoulder is moving forward - light pinkie. That results in her moving her front feet a little less forward and reconnects her with me. If it is more touch & go, a tug on the rein farthest from whatever has her concerned tells her to forget about it and let ME worry about it. A sidepull doesn't allow either of those techniques to work with her.
2 - If she is galloping and doesn't want to stop, then a bit gives me more leverage on her nose than a sidepull does. She sticks her nose out and has her head almost parallel to the ground, and a sidepull will pull up on her face farther than the corners of her mouth, so it simply has less mechanical leverage to bend her neck and get her to yield and slow. A snaffle bit also has some problems, since it slides back against her clenched teeth when her head is like that. I've ordered a Kimberwick (spelling?) to see if a mild curb will help her to lower her nose. With Mia in a full-bore gallop, lowering her nose is the key to slowing her.
Trooper has never bolted or refused to slow from a canter or gallop, but he acts more confident with a bit. But since he hasn't ever bolted, I simply don't know. Lilly never bolted on me either.
I haven't tried bitless with Cowboy. He is usually ridden by my 100 lb daughter-in-law, and I don't think she has the upper body strength to force the issue with him bitless. He only bolts in an arena, and responds well to a one-rein stop. He's a BLM mustang pony, and he believes in being careful on the trail.
Very old photo, from shortly after I started riding, but it shows the bitless halter I used with Mia. Still use it as her daily halter. I think you can see how this design allow zero leverage on the nose once she stretches out in a run:
Same design on Trooper. This was on my first-ever attempt at cantering. It says a lot about their differences that I felt more comfortable, for my first ever canter, to ride Trooper in a sidepull halter. But again, it shows how little mechanical advantage THIS design gives. Notice where the reins actually connect next to his head. And no riding critique, please, since this was a couple of minutes into my first attempt at a canter. It was also Trooper's first canter in over a year: