I've only got a gas grill at the moment (condo, no way to deal with charcoal) but I've used a pizza stone on the grill several times.
My way around the whole burnt crust/uncooked toppings has been to let the stone come up to a lower temp, say grill on 50-60% power. Then put the pizza on the stone, and let the grill rip wide open. Seems like the lower temp stone/higher air temps cooks everything nicely.