I have had success teaching the deep water start, with one key idea. Keep enough of the ski out of the water so that the front toes are nearly at the water's surface. This starts the ski planing very quickly and prevents them from being pulled over the front of the ski.
Also, I think it is really important to break the start up into two stages. 1) skier tells the driver "in gear"...drag them a little bit until they have their balance sorted out. 2) second command is "hit it", they will pop up from being dragged almost instantly with a gentle progression of the throttle
Beginners can't handle a hot start, or even a medium one...a nice gentle progression on the throttle.
The other keys are knee to chest, arms straight and don't stand up until you are past the spray.
Good luck...your kid will get it soon!