Loosie - I love the Dr. Cook's bitless too - but I've found that it applies a lot of pressure to a horse's poll, most of my horses are trained that poll pressure means put your head down (not like a false frame, but more to help shift them out of flight mode when I'm leading if needed) but horses who have never experienced poll pressure will often have their head in the air avoiding it, or be tossing their head around alot. It's a fantastic design for horses who know how to interpret the cues, but because it uses feels so different from a typical bridle I feel few people really know how to train a horse to be comfortable in one. Except one therapy barn I worked at, all haflingers, all in Dr. Cook's. All responded SO well and they didn't have to worry about the students being too rough. The only other thing I find, is that the release of pressure isn't very immediate on the Dr. Cook's, which is fine for a trained horse who doesn't need and extreme obvious release to say 'you did it right' but not great for starting a horse in.
I really like the Indian hackamore because it usess the same cross under principle as the Dr. Cook's but it only effects the nose area. I also find with that the release is slow unless you have the type with the metal rings and good quality rope, some ropes are sticky.
I also believe the the top of a horse's nose can be more sensitive than the inside of their mouth, there's only a thin layer of skin there and a number of sensitive pressure points. While a horse's tongue is a very strong muscle, capable of absorbing some of the pressure. So there are many bits that horses would prefer to bitless. Look for mullen mouth bits, or Myler bits, those all have a great deal of tongue relief which is what most horses fuss about.
I also agree about getting the horse's mouth/teeth checked.
I also agree with loosie about avoiding the Mechanical sorts, those don't have very clear communication but they can be very intense, I watched a small woman break a big bull-dog quarter horse's nose with an ill-fitted mechanical hackamore. I would also avoid rigid rawhide bosals, they are just very strong if you want to neck rein try with something less strong first.
I like the designs of most sidepulls and x-overs barring the things I've already mentioned.