Yes, it does. When you pull the rein, it rocks the shank backward, but since the chain is attached ABOVE the pivot point, (which is the mouthpiece, itself), then the chain experiences a pull toward the horse's lower jaw. It pinches the lower jaw between the bit mouthpiece and the chain. The pressure can get really intense , even to the point of breaking a horse's jaw, if the shanks are long ( longer shanks = more leverage) and a person was to pull horrible hard, or other unexpected circumstances. Mostly, people who ride with shaanked bits have the horse well trained to respond to light pressure, and the riders have their hands trained to be very careful in applying pressure.
No true snaffle bit has any leverage at all. A real snaffle never uses that chain.
You will , however, sometimes see people using a "chin strap " (vs the chain, which is called a "curb strap"). This chin strap has not pull on it at all. It is only there to help prevent the bit from pulling through the horse's mouth when the rider pulls hard on ONE rein, sideways.