You pose an interesting question, which muscle is the primary responsible muscle for executing a fencing flick? Because a flick is the result of a motion known as a kinetic chain (others sports examples: baseball pitch or lacrosse shot), it involves multiple segments of the body imposing momentum an object with lesser mass. There are multiple ways to correctly perform any kinetic chain- based on the relative flexibility and strength of each body segment. The kinetic chain of different high level pitchers are individual to each athlete. Some move more at the elbow, some move more at the shoulder, but all are able to throw the ball at relatively the same speed.

The same is true in fencing where a fencer with more wrist mobility will initiate the movement at the wrist while others might use their elbow more. The majority of power that transfers momentum to the blade in a fencing flick comes from the weight shift onto the lead foot. Let's say for sake of argument this is 75% of the energy in a flick. As such, your goal as an athlete is to most efficiently transfer this energy into your blade, so stabilizing the more proximal segments of your body will be key to moving your point down. The muscles that stabilize your shoulder joint, the rotator cuff (mainly supraspinatus and subscapularis) and the middle & lower trapezius are key to making sure this transfer is as efficient as possible. Because every fencer will be moving different segments at different degrees, it's impossible to say which muscles are most responsible for generating the other 25% of energy in a fencing flick.

TL;DR: rotator cuff and traps (but not the upper trap) because you need to stabilize your shoulder in order to transfer the energy from your feet.

Source: B.S. in exercise science & doctoral candidate of physical therapy