There is a rodeo clown out there selling halters and calling them natural horsemanship hackamores, but they aren't. They are just overpriced rope halters with a mecate rein attached. You could put a cat in the oven and call it a biscuit, but it's still a cat.
A rope halter can be a great tool for casual riding, for practicing your communication skills with your horse, for almost any riding that doesn't have rules regarding required tack. Don't delude yourself that it's oh-so-gentle just because a rodeo clown slapped the name 'natural' on it. Those nose knots can be very harsh, especially when you're pulling back on reins or snapping/jerking the rope toward the horse (putting life into it, as the clown couple likes to say).
I make my own rope halters and put a leather pad under the nose band and knots to make it less harsh on my good horse. I also ride her with a flat webbing halter sometimes. I use a contest-type rein with snaps on both ends, so my riding halter is also my leading/tying halter. I just snap one rein into each of the loops under the chin. Because I make them custom, they're not sloppy-floppy loose, which I don't like whether leading or riding.
A bosal can be milder than a rope halter, or downright vicious, depending on how thin, stiff, and heavy it is and the texture of the covering. Using a bosal requires a different reining technique that I won't go into here. They were originally inended to be a step on the way to a finished spade-bit bridle horse, but they can also be a fine bridle for daily use, if you're not trying to follow in the Californios' footsteps.
I disagree that a snaffle or any bit provides more control. Control comes from training, not from tack. I will say that a snaffle can provide a "more precise communication* than a rope halter. I know of horses that are easily controlled when ridden with a halter, but they prefer the cleaner, crisper communication of a snaffle.
A sidepull is another bitless option. It offers a less 'sloppy' communication than a halter and it can be rather strong, depending on the type of rope on the nose and if it's padded. I consider it the snaffle of bitless, because its action is identical to a snaffle, only on the outside of the horse's head.
The Dr. Cook's Bitless Bridles that I've seen don't give a quick enough release to suit me, unless they're made of nylon webbing. Leather and biothane beta tend to hang up in the rings or drag against them. Since I don't like the look or feel of nylon bridles, I don't care for the DCBB. I've heard of horses that have a strong dislike for the 'head hugging' action, too. Looking at it in the tack shop and the price tag were enough to keep me from buying one, but I made some variations on the idea and none became my favorite bridle.
A mechanical hackamore is extremely harsh and a poor communication tool - but I know you weren't thinking of trying one of those. Just throwing it out there for comparison.
I've experimented with every headgear mentioned here except the DCBB, on my own and client horses.
I like to use different gear for different activities. Only have 30 minutes to hit the trail? Put on my helmet and snap reins to the halter and let's go, Pony Girl!
Chasing cows or cross-country jumping, or a guest rider? The flexible, smooth bosal is just right for helping her keep her mind on her work when she's a little excited, and for strangers to manage her without getting in her mouth.
Dressage lessons? Snaffle, of course, and I prefer a French link.
I personally enjoy trying different bridles to see how they work and how my horses respond to them. You might find something that you and your horse like better than what you're using now.