Peter, as always, beautiful looking pies.
Pizza bakes with stored heat, not just the heat on the surface of the stone. While using the broiler to pre-heat the stone might drive up the surface temp, it doesn't drive up the core temp, and the core temp is where it really matters. I'm not telling you to nix the broiler re-heat- it does trim a little from the bake time, but the 615 benchmark that you hit is not a realistic representation of what the oven is capable of. If I had to guess, I'd say that the broiler gave you an effective bump of about 5 degrees, putting you at 585.
My feelings are shifting a bit on high alumina kiln shelves. There are no regulations governing how the shelves are labeled or what they can contain. I'm not talking about safety. Kiln shelves are safe to bake on. But the term 'high alumina' doesn't necessarily mean much, nor do the other claims on the Bailey website. Someone definitely needs to, at some point, test the Bailey shelves, but between the fact that they haven't been proven and your low-ish 585 peak temp, I don't think that person should be you. It's too much of a crapshoot. Mike's SF shelf is a proven performer. I can't guarantee you how much time it's going to trim, but at least, unlike Bailey, it has a track record. If you want to go the ceramic route, that's the stone I'd go with. If, on the other hand, you want to guarantee a 4 minute bake, then I'd suggest steel. One nice thing about steel is that, with your 585 peak temp, you don't have to buy as heavy of a stone as other members here have. You should be able to get away with 3/8" steel.
All THAT being said
another thing to consider is that there may not be a reason to trim any more time off the bake. I generally push people towards 4 minute bakes when their oven setup is producing 7+ minute pies. 5 minute pies, though, that look as good as yours? If you're curious about a slightly shorter bake time, sure, make the investment, but you might not see a vast improvement. Also, if crispiness is important you, then 4 might not even be the right direction, as the crust will soften up as you decrease the bake time.