Hitting jump shots cutting off of screens is definitely more difficult for most players than sinking them when their feet are set upon the catch. I agree that Trevor plays too fast at times. He's impatient when reading screens, he rushes to get shots off, and he often speeds through his shot mechanics. As you point out, hurrying as he cuts away from the basket off of these pin-sown screens leads him to fade or lean as he shoots because he is off-balance. He is not a good enough shooter to compensate for poor mechanics.
The real key is for the player to set his feet before receiving the ball. Cooney is not great at this. Also, he doesn't curl the screen or fade off (bump) of the screen well. When a player cuts off of a screen the same way no matter how the defense plays it, he is wasting the screen. Occasionally, Trevor curls the screen (once every few games), which usually leads to good things in the lane. However, he does this far less often than he should. Furthermore, I can't remember the last time I saw him bump a screen. Andy Rautins executed this move well by his senior season. Even when he correctly cuts straight off a screen, Trevor still needs to use shot and pass fakes more to create space for his shots.
The good news is that Trevor has shown some growth in his game this season. His handle has improved, and he seems more confident when taking the ball to the hoop. He has also improved his passing some. If he can keep developing his fundamental technical and tactical skills, he could be a valuable piece to a solid tournament team this year and next.
In terms of the offense as a whole, getting more looks at the hoop with our shooters' feet set on the catch would help. This could be done in transition, but it can also be done through half-court offensive design. Coach Boeheim loves two types of screens in his set-play offense: pin-down screens and ball screens. He also has players execute a stagger screen for the best shooter in the "Double Fist" play, and will occasionally work in a cross screen to isolate a player on one side of the floor ("53" and "54" plays). That's about it, which is part of the shooting problem. We rarely use fade screens, zipper screens, UCLA screens, flex screens, etc. We don't even use dribble handoffs as much as we used to when Gerry McNamara was running the show.
In short, more player and ball movement would help players get more looks moving toward the basket or with their feet set prior to catching the ball. Moving the help defense allows for driving lanes, which should create layups and kick-outs. Additionally, fighting through screen after screen takes a toll on a defense over time, causing slower closeouts on shooters and more fouls. Finally, this movement would hurt defenses that want to double Christmas in the post.
That said, it doesn't matter how many open looks bad shooters get; bad shooting is bad. Only a couple of the players look like "true shooters"--McCullough and BJ, for example (yes, BJ. Each shot he takes looks good, even though he hasn't hit many outside jumpers yet. The young man also understands the fundamental tactical reads in an offense, sliding to the correct areas based on ball position); the rest of them look like they can occasionally hit shots. This is not a recipe for consistent offensive success, regardless of offensive scheme.