So if the horse is THAT sensitive to what's going on around him and touching his body, why can't he feel a subtle change in the tension of the strap running down his face and under his chin?
He can. I have some opinions about these things, but they are just my observations.
First, I believe the most important thing is how any bit or bitless feels to the horse. I've read article after article on the concept of different bits and how they work. Yet it is always the horse that tells me if it is harsh or gentle to him. Some that are supposed to be very gentle seem harsh to the horse. Some that are supposed to be severe feel gentle to the horse.
Bits also are only a tool, and the most important key is the hands behind the bit. A rider with good hands and an understanding of the bit's action can use most bits in a way that the horse will accept, at least short term. But that may not be the best bit for the horse, and a sensitive horse might hate the bit at the outset.
said, the biggest problem I've seen with a Tom Thumb is that it is often used by beginner riders with insecure seats, and they have no clue about how it works. They just get on and ride, and end up using it harshly. That can happen with almost any bit, but a snaffle will be more forgiving.
I don't think the main issue with the crossunder is that it doesn't release quickly. The pressure releases the instant your hand stops pulling. It doesn't clamp down on the horse's face without any rein contact. It doesn't drop away in a big release, but the pressure is provided by the hand on the rein, so without contact it is no more pressure than wearing a bridle on the face.
However, I've ridden some horses that don't like the squeeze around the face in the first place. So it's not about how fast you release, but rather they don't accept that contact in the first place. If it doesn't make the horse too claustrophobic, the horse can learn what that squeeze signal means. But then many horses realize how gentle that pressure is, and may ignore it when excited. Others love it and respond well. I've ridden some OTTBs in them and they went very nicely. One of my Arabs can't take the squeeze without panic. The other one doesn't think there is enough of a signal to respond to.
Once I thought Tom Thumbs were evil, but I've since seen them used on horses that accept them fine. The horse has to be a little more balanced, and able to travel without contact. The bit has to fit, and the rider has to use it only for signals and with a clear release. The horse at my barn that goes well in one travels very round and balanced on his own, and mainly on a loose rein. The rider neck reins, and uses direct rein signals for fast turns with a good release. No head tossing and the horse is happy.
I wonder if the idea that they poke the side of the horse's face when direct reining is actually a perception of an issue I've seen with jointed mouthpieces and curb chains in general. When the bit breaks due to the rider pulling on one rein, the chain can pinch or jab the horse on the side of the pull. So instead of getting a nice cue to turn to the right, the horse instead is getting poked on the right, which makes him want to not move into that pressure. This can be eliminated by using a curb strap that does not have the chain start right at the bit cheekpiece. Such as the ones with leather, nylon or beta straps connecting them to the bit.
Not all horses have this issue, but horses have a variety of head types that make the curb fall in different places on their chin or jaw.