Through all this you have made me think of trying the broiler in my oven in lieu of the bake mode. This oven says you can broil all day at 550 with the door closed. So what I'm thinking is let it soak with the door closed heating the stone, then shovel in the pie where the stone works on the bottom and the broil element does the top. Can't hurt to try...
John, I'm not a big fan of broiler pre-heats or bake pre-heats with last minute broiler 'bumps'. Because your broiler is so close to the thermostat, it's driving up the temp of the thermostat quickly and shutting off. It's not a soaking heat from a bake pre-heat, but more of an intermittent blast that might drive up the top of the stone a bit, but doesn't really penetrate.
Fast baked pizza is not about the temp that's on the top of the stone, but the temp of the core. A 1/2" stone that's 600 degrees throughout will bake a pizza far faster than the same stone with 1/16" of it at 800 while the rest of it is 500.
That being said, even if you could get the heat to penetrate, the broiler answers to the same traffic warden thermostat that the baking element answers to. Peak oven temp is peak oven temp. That thermostat is going to cut out at, in your case, 600ish, regardless of whether the broiler is on or the bake element is on. All you're doing with a broiler pre-heat is extending your pre-heat time because of the relatively ineffectiveness of IR radiation compared to convection.
Now, I don't think the numbers are high, but there are some oven owners with broilers that don't shut off. I don't comprehend how a broiler that doesn't shut off could possibly be safe, but, since I don't hear about houses burning down from this sort of thing, I can only assume that, for these handful of people, these perpetual broilers don't carry any risk.
These kinds of ovens are definitely not the norm, though, and I think it's more of an older oven phenomenon than a newer one. I don't think you're one of these lucky(?) owners, but you might as well crank your oven to the highest broiler setting and see if it stays on. If it does, then there's no need for a new stone.