I love bit-topics. Seriously. LOVE.
First, some clarifications... A Tom Thumb is not a snaffle. A snaffle is defined as a non-leverage bit. Anything with shanks or any bit where the reins do not attach at the mouth is a leverage bit. The tom thumb you posted a picture of is a JOINTED curb bit (a bit with shanks and a jointed mouthpiece). The way a bit with a jointed mouthpiece, snaffle or leverage bit, works is when pressure is applied to the reins the bit pinches the tongue and pokes the roof of the mouth, creating discomfort when the horse fights the pressure. The horse can only move comfortably and breath comfortably when the pressure on the tongue is released through submission. There are different levels of severity with jointed mouth pieces... the loose-ring being the most gentle, the shanked bits being the harshest.
In terms of mouth pieces, a double jointed bit is really the softest way to go. Mullen mouth (solid piece) bits are kind too. They do not pinch and rely soley on pressure at the pole for control, not pressure on the tongue. They're great for a horse who moves well on a loose rein in a jointed bit.
If you use direct rein, it is best to use a snaffle (non-leverage). If you horse really goes well in a curb bit, I HIGHLY recommend a Kimberwicke (pictured below). They are very mild curb bits. A kimberwicke with a mullen-mouth (straight bar across) is EXCELLENT and great for direct (plow) reining.
If you use indirect rein and your horse actually KNOWS how to neck rein (I'm not talking about moving your hands far enough over to one side that the bit gets contact... I'm talking about responding to light pressure from the reins on his neck), then tom thumbs are fine because you hardly use them except when you need them. If you have a tendancy to be heavy-handed or if your horse is hard-mouthed, these are not the bit for you. A stronger horse doesn't need a stronger bit, it needs the right bit. If your horse likes mouth contact, I will swear by a Kimberwicke. You can use direct rein when needed, you can ride it as a regular snaffle if you want, but you can attach the reins to the lower slot and it works like a low-shank curb
This is primarily considered an "english" bit but honestly, it doesn't matter what discipline you ride as long as you're using the right bit for you and your horse. This bit is used in dressage. Anyone who has seen dressage knows that these rides need quiet hands and maximum control. This bit allows quiet bit contact with mild leverage action for a great natural headset.
The bit pictured above is the mildest leverage bit you'll find. It provides maximum tongue relief, unlike a jointed mouthpiece. You can get jointed kimberwickes if you find your horse needs a little more, or you can get one with less of a port if you don't need the joint but you need more than the one pictured above.
The reason Tom Thumbs get a bad rep is because a lot of people think "western riding, western saddle, western bit!" But just because a bit has shanks and a curb chain doesn't mean it'll work for you. Bits aren't "one size fits all" so to speak... you should try plenty of things before you settle.
One last thing on parting: you get what you pay for. Herm Sprenger is a company that makes ergonomically shaped bits (WAY more comfortable for your horse, based on an extensive study on the shape and workings of a horse's mouth). They cost well over $150 but they will last you forever and you will notice the difference vs the $10 pot-metal bits at Tractor Supply that chip and flake in your horse's mouth and cut his lips.
Be kind to your horse!