All right. Let's discuss the topic of Tom Thumbs and Argentine Snaffles, a surprisingly controversial subject… First off, when I say "snaffle mouth," I mean two-piece, single-jointed. "Snaffle" is a misnomer, but a commonly accepted one, quoted by big name trainers in their videos, written about in magazines, and sold in tack catalogs. The word snaffle really means a bit with no shanks and direct rein -> mouthpiece action (something that Tom Thumbs and Argentine Snaffles clearly are not), but in the vernacular it often refers to the type of mouthpiece shown in this video, regardless of cheek style. The Tom Thumb is a bit that gets a bad rap for being very harsh--stories have been passed around of people breaking their horse's jaw with it. Now, that seems a little exaggerated to me (I can't imagine how much pressure they'd have to be exerting on the reins to cause that much damage), and while I don't consider the TT to be particularly cruel, I just plain don't like it for a variety of reasons. Many people do use and love this bit, swearing by it and using it for training colts. To each his own, but my personal opinion is that there are far better bits on the market.
Pictured below is a true Tom Thumb bit, as the black show bit used in the video, as mentioned, is not entirely a TT due to its curved shanks and slightly higher degree of freedom around the mouthpiece/shank connection.
That, right there, is a true Tom Thumb--and the worst one in existence. The straight shanks don't give the horse any warning at all--the second you apply pressure, they pop right around, snatching the curb and activating the tongue-pinching and bar-crushing effects with the mouthpiece. It also makes direct reining difficult--the straight shanks, combined with the way the mouthpiece attaches to the shanks (no freedom whatsoever), mean that when you pull on one side, the entire bit twists rather painfully in the horse's mouth. The curved shank versions aren't quite as bad, but it's still quite difficult for the rider to convey the intended cue to the horse, no matter how light or heavy his or her hands are. This bit just doesn't have a whole lot of finesse in the horse's mouth--it's all or nothing. And any efforts at direct reining will likely result in confusion for the horse. While many horses do ride well in a TT, this often has far more to do with the kindness of the horse's temperament than the quality of the bit.Now compare the above bit to this, your standard Argentine Snaffle.
At first glance, they look very similar. But the Argentine is much better balanced. Look at the curved shanks--these make it less harsh, and a slight touch of the rein will give a horse warning that a cue is coming, giving the horse time to react. Also, the joint between the two parts of the mouthpiece is finished better, so it will be less likely to pinch. Most importantly, the attachment from mouthpiece/shank is not fixed in the vertical direction. This gives both parts some freedom and independent movement. Slight jingles in the reins are often all that is needed to give a cue and thus get the desired response from the horse without the yanking that is often required with a Tom Thumb. Ask any horse--most will respond so much better to bits of this type.
In regards to folding and pinching of the mouthpiece into a palate-poking "V," unfortunately I was not able to give a good visual demonstration of this effect. Keep in mind, as stated, that the bit will behave slightly differently in the organic environment of the horse's mouth, interacting with tongue, bars, lips, jaw, and so on. The movement of the horse's mouth certainly has the ability to change the orientation of the bit, and this could lead to some pinching with any mouthpiece, but more so with a two-piece "snaffle" mouth like this. However, in a curb bit like an Argentine or a TT, when the reins are held two-handed for direct-reining, the hands are wider apart than the cheeks of the bit (and the cheeks are further held apart by the very size and structure of the horse's mouth), so the bit will not fold in on itself. Even when riding with one hand, the reins are separated by the width of the horse's neck, and thus greatly minimize the inward force pulling the shanks together, and thus folding the bit. Pulling directly back, harshly, while riding one-handed with a centrally-positioned rein could perhaps begin to crush the bit inwards, but this is very poor riding and should be avoided regardless of the choice of bit. When neck-reining (and even when direct-reining), too, the primary control of speed, stop/go, and turning should come from the rider's seat and legs and the "push" of the reins across the neck—not from bit pressure, again regardless of bit choice.
When are these bits appropriate? In my opinion, the answer for the Tom Thumb is "never." For the Argentine Snaffle, I find it to be a good transition between a regular, true snaffle bit (O-ring, D-ring, etc.) and a curb of any kind. The mouthpiece is familiar to the horse, the shanks are short and relatively mild, and well made bits allow for some degree of independent side action, meaning that it can be used for direct-reining, flexion, lateral movement and so on. Recall again that bit quality is of the utmost importance. Some bits marketed as Argentines are really nothing but glorified Tom Thumbs. Try the bit out first to make sure that it works the way you think it will. The Argentine may also be a good choice for trail riding (so long as the hands using it understand its action and can use it appropriately) and for training any sort of horse who appreciates a bit with this design—and many do.
As with any bit, both of these require an educated hand with good "feel." Whenever selecting a bit, play with it in your hands first to get an idea of how it works. When riding, visualize the bit in the horse's mouth. With practice, you can even begin to feel what it is doing—you can actually "see" the rotation of the independent sizes of the mouthpiece, feel the curb chain activate, and so on. Thus you can temper your rein motions accordingly to transmit the clearest—softest and most precise—cue to your horse.