If you ask those same people to hit as hard as they can and not worry about where the ball goes most still start the motion with their arms with none of the preliminary body movement that is observed with the proficient. I just went through this in the last couple of years with my son and tennis. As my son hit HS and went out for the tennis team, I was annoyed to find that in years of tennis lessons this basic fundamental (and others) had not been taught (but not nearly as annoyed as his frined's parents should be who were springing for private lessons). A few weeks later (I taught him 5 days a week and yes, 30 years ago I played tennis a lot) he was much better but still not able to generate the power he could with better use of his body.
Once the skills are learned you no longer think of the body movement just of what you intend to do. Tiger (normally) does not think of how to swing but of the shot he wants to hit (as he's said in a number of interviews). However it is clear that when things go wrong he does think about his swing and not just the shot.
BTW - Any drills mentioned here that I've created don't develop proper body movement. The Bertholy drills do.