Yeah some people prefer to rock more forward vs pressing more up from the ground. I press more forward from the ground when I want to keep the putt lower and press more straight up from the ground when I want more loft or in between the two for a balance of both.
However, the main point I was trying to make is that since your arm is attached to your body, whatever body momentum you create by pressing from the ground, the arm gains that momentum. So if you are going to lift your arm, waiting until the lower body breaks the inertia and gets the body up to speed makes it a smoother and more effortless motion.
Also, of course as we all know different cues work for different people. Instructing someone to try to create a 'wave of momentum' from the ground up may help someone a lot, or explaining it more granularly as multiple separate motions blended together in a sequence might click more.
However, I also think trying to instruct someone to do a 'wave of momentum' / fluid motion is often premature if they can't even perform simpler motions smoothly with coordination. Such as, if they have very poor control over their wrist, I think addressing that first separately will make the fluid motion instruction that comes after much more fruitful--but in this case it might just be best to have them think about not using their wrist at all and add it back in later once everything else is working well together.