I think every captain I've dived with turns off the engines when boarding divers. My concern is not that the engine is on, it's that if the captain had to move the boat for some other emergency (another boat approaching at speed, an injured diver brought to the surface at some distance, etc.) he would likely clear the area around him in a hurry, fire up the engine and off he goes, with you "standing" on the bottom of the boat with no fins on. That little joke could turn tragic before you could do anything.
I am happy that the op I use in Cozumel would frown on doing this. Diving is risky enough (that's why we like it), but adding that activity to the mix seems unnecessary. Looks like I'm in the minority though, I'm surprised at how popular it is to do.