In teaching beginners to flycast, I've stopped talking about how to move (or not move) wrists, elbows, and shoulders. Instead I just focus them on the goal--a short, relatively flat stroke that accelerates quickly and comes to a sudden stop, sending the line straight out behind them, and then a similar, short, relatively flat stroke in the opposite direction that accelerates and comes to a quick, firm stop, sending the line straight out in front of them. I find that different casters use more or less wrist and do different things with their elbows and shoulders depending on what's most comfortable for them. So trying to be too prescriptive about how to move body parts during the cast isn't always helpful, since different people will find different casting movements most comfortable. Also, it seems distracting for beginners to focus too much on how their body is moving, rather than focusing on the rod and line movement they are trying to produce via their body movement.
As a more accomplished caster my biggest bad habit is rotating my shoulder forward and leaning into my forward cast, rather than standing straight and letting the rod do the work.