I (and wife and daughter) ride our horses in halters on occasion - just to "change it up" for the horse and ourselves.
I also start my horses in a halter or hack...depends on the horse and then transition to a snaffle after a few days.
However, my horses are either rope horses, reining/cow work horses, or drill team horses....and although they can be ridden without a bit, it is not wise to do so -- no matter how well trained and "soft" the horse is....when you are working at a full run, chasing, going in and out of other horses in a figure 8 at full speed carrying a rope/flag etc. you better have a bit in case of emergency.
When you are riding one-handed you need to be able to fully control the horse with as little effort as possible. At times, even the most level headed horse will become excited --- ie - the cow won't turn on the fence and wants to run at the horse - you had better be able to move the horse asap, and in the right direction/angle, or you will not have a pleasant ending....or, when roping, the cow turns in front of your horse...you need an instant response(and, remember you only have a left hand for cues) to keep your horse from getting tripped by the cows back feet....
Now, having said that, if you can take your horse, have 500 folks in the stands, music blaring, flags flying, signs flapping and run your horse 5-10 feet from the fence turn him into the fence, stop him, and then circle him to the right and then take off again down the fence in the opposite direction(on the first stride away from the fence) using nothing but your left hand with a halter...then you might be able to do that after you insert the cow into the equation....but, I wouldn't want to have to rely on that.
And there are many other instances when riding in a halter would not be the "preferred" method.