sorry there is one exception : The only diagonal gait of the gaited horses is the Fox trot. All other gaited horses are lateral. And yes a foxtrotter could have a lower head set, because it is half way trotting.
If you have had good luck with a regular snaffle, that's fine, your one of the very few. OR, you did a whole lot of inappropriate training. In either case, there are a few it will work with. In general, that is not the case. Most of them, snaffle or otherwise, need a "comfort" bit, to prevent pinching and unnecessary discomfort. If you have not checked your horse's mouth for fit, how do you know whether it fits or not? I've seen a whole bunch of trainers that simply use the same bit on all of them, regardless of the fit. Just because you have always used a regular snaffle on a horse, does NOT mean it is a good fit. Most horses are very adaptive and will learn to accept a lot of poor fits. BUT, just imagine how much more you could get them to do freely, if you avoid uncomfy bits, etc.
In any case, the point I was trying to make about the snaffle, is simply, you can not assume a snaffle is easy on the horse without first checking. A snaffle can be just as severe as anyother bit, if it does not fit. I think the point went over the top.
Indeed you can injure a horse quite badly with an ill-choosen or ill-used snaffle bit. But the problem is not the "snaffle" action (which is direct) but in selection of the mouthpiece. On my bit wall I've got 9 different snaffle bits and seven different types of mouthpieces are represented. What I choose with any one horse will depend on what I want to do with them. Or if I'm working on correcting a problem what that problem might be.
You are correct that many people make false assumptions about snaffle bits. One of the most common is that "a fat bit is less severe than thin one." This true if both bits fit correctly. But I was at a trail ride and saw a horse being ridden with a hugely fat snaffle; it made the horse look like it had a mouth full of bubble gum. The horse was quite uncomfortable, on the edge of being fractious. The owner was clueless. Ron White's famous observation was again verified.
Amongst the folks I work with the snaffle is by far the most common bit used. When I visited Brazil it was the most common bit used (there are 350,000 Marchadors in Brazil). I don't find my experiences at all unique.