Not an instructor - but I have worked on the range with a swing speed radar and got some pretty objective feedback on what increases my clubhead speed through the ball.
I would say that, by far, the biggest jump I got in speed came - not from coil, or weight shift, or width of arc, or X or Y factor, or extension or connection or release or lag - but simply from upping the overall tempo of the swing. And the most obvious increase in tempo seemed to me to be on the backswing.
I also felt that the extra speed was initiated through my arm swing, rather than my core turn - though clearly the 2 need to move in synch.