I read an excellent piece on this in Mark Rashid's book "Life Lessons From a Ranch Horse" (I highly recommend this book btw!).
He talked about how at one of his clinics, he was warned about a woman who was a diehard NH fan was there who liked to accost clinicians about why they used anything more than a rope halter. She would hold up entire sessions for half an hour or more arguing with the clinician during Q&A time. Mark was prepared when her questions came and he used the example of the horse he'd just worked with, who had come in confused and frustrated after being in only a rope halter- he even had rope burns on his face from pulling against it. When the horse was put into a snaffle, he was able to understand what his rider wanted and comply with the request, so it helped him become more relaxed and focused.
In short, I believe that each type has it's place and it's time when it will do the most good. I consider it to be another tool in the toolbox, the difference between a Phillips' Head and a flat-head screwdriver. They both do the same job, but one will usually work better than the other in a given circumstance.
That said, I *DO* believe many riders could benefit from learning to ride bitless, especially at first when they are developing their hands and seat. I think a lot of riders are trained to rely on a bit like the brakes in a car, when they need to be learning how to communicate with their seat and legs. To me it feels a little like handing a 5th-grader a hammer and telling him to go build a house. He might be able to figure something out, but the result is rough and certainly ain't going to be pretty.