It's interesting you mention that (re: the hydration.) I've been making pizza for maybe 6 mos now (I'd say 10 times over 6 mos but have been reading about it obsessively.) One of my initial attempts involved the recipe for Lahey's Co. pies, which clock in at around 75%. I could barely shape it, let alone get it off the peel. (Ordered delivery that night.) But my past two attempts have been right around 70%, and the dough has been extremely manageable. I have to attribute that to working the dough, vs Lahey's no-knead method.
I'm using a KA and start off with a 20 minute autolyse using maybe 85% of the flour. Then 6 minutes on the stir setting. But as soon as I start the mixer, I add another 40g or so of additional water. This basically turns it into a soupy mess. But from what I gather, this is really when a lot of the gluten development takes place. (Varasano speaks to this.) After 6 minutes or so, I might take the speed up to 1. That's when I start adding the remaining flour. Maybe mix for another 4-6 minutes, adding enough flour so that it never quite balls up but gets close. A 20 minute rest before balling, an additional 10 minute rest, and then into the fridge. The resulting dough, even at 73%, is a dream to handle. No rips, tears, extremely pliable. No sticking. I've stopped using semolina on the peel... just straight AP.
Anyway, to the rest of your post, that's exactly what I was planning to try next time. My only concern is that my broiler cycles (I assume it hits the oven's max temp, turns off, then cycles back on), so I'm curious as to how hot the stone will get. I imagine pretty hot, though. It should absorb a fair amount of heat. If not, I'm tempted to heat the stone to 700 or so on the grill, as I've been doing, then transferring it to the oven, directly under the hot broiler, before making the pie. I imagine I could do a 2-minute pie, or close to it, if I timed the broiler properly.
Either way, I'll report back.