Accelleration is the key and it’s done with the hand. The hand goes back and the hand returns, that ‘s all there is to it. Thinking about the shoulder and arm is something I would strongly advise NOT to do.
The Game is played in the Hand/Wrist/Forearm, and that’s where to concentrate for the quickest improvement. Remember, the “moment of truth,” is at impact and “whitey never lies,” so keep it simple and precise. Pool’s a game of extreme accuracy and hand/eye coordination, not arm/eye coordination.
The follow through happens naturally, I merely suggest not trying to extend it OR restrict it. A nice pause before the transition from backstroke to fore stroke is an effective way to control tempo and follow through.
The faster you try to take the cue back the more energy it takes to redirect it. This also leads to “jumping on the ball,” trying to gain power in your stoke from anything other than a controlled delivery.