Stone too hot is the obvious short answer. Everything I learned when I got really into this (during early covid) is that the grill is really the wrong device for cooking pizza. It has more to do with the shape and all the room on top and heat escaping from the grill than the stone. It's pretty well documented if you research proper techniques for cooking pizza.
To get it right, the pizza should be cooked very hot and fast. Just ball parking numbers by memory here (because I'm too lazy to look it up), but a 700 degree cook should be done in about 3-5 minutes or so. But that's with 700 degrees everywhere, not just on the stone. That's why pizza ovens and regular ovens work better. You load from the side and the temps on both the bottom and top (called the lick in a real pizza oven) are pretty close to one another.
So, the stone at 650-700 degrees was probably close to the right temp, but the rest of the grill will never be close to that given that you have to open it up to put the pie in, check on it / turn it, and take it out. You lose that ambient heat really drastically. That's why you had to cook it 12 mins to get the toppings cooked, which is way too long for that hot of a stone. I'd bet if your stone was 700 degrees and your grill was around 550 at its peak during the cook, the ambient heat of the grill was actually averaging somewhere around 300 with opening and closing throughout, plus that heat was never very close to the toppings, while the stone probably held closer to 700 throughout. That's a huge temperature discrepancy.
So, the problem isn't technically your stone temp, well it is, but it's a little more complicated than that. The problem is the rest of the temp. To get it right, you need to figure out how to keep both temps closer to one another. Probably the only way to accomplish that on a regular grill is to get a kit that holds heat in more like a pizza oven, something like this...
That will get the top heat and the stone closer to one another and keep that heat close to your topping, which is correct. Then you can cook the whole thing in around 5 minutes with probably one turn, as that front part will be cooler than the back, so the turn is needed for even heat distribution.
Otherwise, you need to have your stone cooler by a few hundred degrees, then you can cook it closer to your 12 minutes, which isn't as desirable.