Let me explain how the drive off the rubber actually occurs. When he is into his leg lift his body or center mass should drift slightly towards home plate, .08 second mark of Greg Maddux video below.
Then he drops down with the back leg- .09 second mark of Greg Maddux video - his knee on the back leg should be rotated out towards the shortstop position or close to it.
The drop of the back leg, combined with the drift, will cause his back leg to automatically begin to drive towards the plate due to gravity. So, at this time, the drive will automatically occur. There is no need to push hard off the rubber. He can drive just as his foot is coming up onto the big toe if he wants, but I personally do not have my pitchers push off the rubber.
As stated previously, pushing hard off the rubber causes the body to rise and the pitching arm to lag behind making it difficult to catch up. This causes major stress on the arm and shoulder and makes for poor control.
So, leg lifts and drift occurs towards home plate, Back knee is rotated outward, Then drop of back leg happens… The back leg will begin to extend outward as if doing a split. .10 second mark of Greg Maddux video. As his back leg extends out hold the torque on the back knee as long as possible.
When his back leg extends, his back foot must roll inwards where the inside of the cleat is almost fully on the ground. .13 second mark of John Lester video above. His hips will continue to rotate, opening the hips completely.
Once his front foot is almost on the ground his hips should be open close to a 45 degree angle, doesn’t have to be exact .13 second mark of the John Lester video above.
His back hip should be pointing between the second base position and the second base just before his front foot touches the ground. For a right handed pitcher his back hip would be pointing somewhere near the shortstop position. His shoulders should remain closed as long as possible as his hips are opening up.
Then after his front foot is completely down his hips will continue to fully rotate open. The back foot will then roll onto the ball / big toe with his momentum pulling the back foot laces down .11 second mark of the Greg Maddux video
You want the hips to open as early as possible, while keeping the shoulders closed as long as possible.
I am using the Greg Maddux video to give a different perspective for you.
I hope this explains how and when the drive off the rubber actually occurs.
If you have any more questions just ask and I will answer them as soon as I can.
P.S. The best way to work on his drift towards home plate is to have him put his pitching shoulder approximately 3-4 inches from a wall as if he is in the stretch position on the mound. Have him fold his arms across his chest, left hand on right shoulder, right on left shoulder. Then have him go through his motion with his leg lift keeping his hands across the chest trying not to hit the wall. He may hit the wall with his knee, shoulder or both the first day or two doing this drill, but that’s okay.
Once he gets to where he is not hitting the wall have him try doing the drill with his full motion using his entire body, arms and legs. Again his arm or hand when extending out may hit the wall doing the drill from the full motion, but that’s okay. He will eventually get it to where his drift will not allow his body to hit the wall. This is a very good drill and he need not rush it.