I won't preheat the stone very long--maybe 4 minutes at most. Now remember, I am using premade crusts like Mama Mia or Boboli and just adding toppings. But just putting down the stone, giving it a dusting of corn meal and then throwing the pizza down has worked for me. Here is a picture of the finished product.
A little late to the topic, but I will chime in as I have been cooking pizzas this way for several years now during the summer. I use my 22" Weber and fill a chimney with briquets, light them off and when they are ready, you spread ALL the briquets along the extreme outside of the bottom grate and put only two or three briquets in the middle of the grate. I am using a large Pampered Chef 15" diameter stone. So the heat has to come up around the edges and the stone isn't taking the brunt of the heat.
I cheat and use premade crusts though. When I am ready to cook I will put down the stone on the grate and sprinkle some corn meal, then place the pizza. All the vents are fully open and this method allows me to leave the top on and watch through the top vent--when the cheese is melted, the crust has just the right amount of 'crisp' to it. It only takes about 8-10 minutes. I usually cook two pizzas, and I will give the second one a new layer of corn meal and the same process.
I've been cooking them like this for years, and my well-seasoned pizza stone shows it!