Bitless & stopping: my experience with my mare.
Mia used to be very spooky. I also rode her bitless only because she was afraid of bits...took me three days of trying the first time I wanted to put a bit in her mouth! That should have been my first sign - or maybe my 4th or 5th - that the "perfect for a beginner rider" advertisement was, shall we say, optimistic?
When a horse bolts in fear, eye-rolling & diarrhea-squirting fear, no bit or bridle will, IMHO, make the horse stop. That first 50 feet to 50 yards, they have to go and pain in the mouth won't stop them.
After that initial bolt, however, I think a bit helps to calm the horse. You can turn a horse bitless. I used a rope sidepull halter, and I could always turn her head. However, that wouldn't calm her. What worked best for me was simply calling her name softly. My mare is a sweet girl, and that would cut thru her fear faster than anything else I tried while bitless.
I eventually admitted I was in over my head and hired a trainer, who concluded that Mia had never been broken to ride. During the 3 years (yes, 3 years!) I had ridden her, she had calmed enough to accept a bit, but she had NO IDEA what a bit meant. So she was started over as a never-broke horse, and part of her training, a big part, was teaching her bit cues. First from the ground, and then from her back.
Once she understood the cues, I found the bit allowed me more subtle cues that in turn helped her stay...well, if not calm, at least not spinning out of control. That was true of my ranch gelding as well. Both horses, when you trot fast or canter, get excited. The mare would excite herself out of control. The gelding would simply hollow his back, raise his head, and go fast with his neck perpendicular to his body.
With both of them, I found it very hard to calm them down bitless. I could turn them in circles and eventually FORCE them to a walk, but I couldn't get them to just ease off and accept a slightly slower and more controlled pace - not bitless.
With a bit, if they start getting excited (which means faster for both of them, until they are strung out), I can milk the reins slightly. Not tell them to slow, but milk each rein in time with the shoulder to tell them 'not so far' with each front leg. What I've found is that both of them, within 6-12 paces, will then slow a little and start to move in a more controlled fashion. Not 'collected', but a tiny bit in that direction.
So for me, with horses that tended to spin up out of control, I found that a bit helps me to calm the horse. Neither bit nor bitless would give me the power to MAKE them stop, but a bit made it easier to settle them down quickly. It also helps me prevent the bolt in the first place, since it allows me to tell my mare, "Trotting is OK, but we're not in a race". For her, that is important.
With a lot of work, both horses are now much calmer. The gelding has turned into an excellent little trail horse. My mare is starting to get the idea. She is by nature a very willing girl, and I may someday be able to return to bitless with her. OTOH, why change what isn't broken?
My recommendation would be to work to teach her the bit is harmless again. It is possible to get a horse to settle after a spook without a bit, but I found it easier to prevent the bolt/settle her after one with a bit.
Old fat guy on a not so spooky mare a few days ago: