Pizza stones are a necessary evil. They can act as heat sinks as much as anything else.
@Againstben, on a grill (of any kind, probably) the challenge is as you've experienced: cook the top without hurting the bottom. Kitchen ovens avoid this because non-broiler heat is always indirect and for a large diameter, flat piece of food that's trouble where heat inherently comes right at it, like on most grilling situations. Even so, when you spread coals around they are still closer to the bottom of the pizza than the (insulated) top, which requires more heat.
On a 22.5" kettle you can place the stone on top of two firebricks laid down on the long end. This raises the pizza up so that the kettle lid can better concentrate heat where it's needed most. I won't claim it's a huge improvement necessarily. Another thing to try is those same 2 firebricks laid down flat. This will almost definitely protect the pizza bottom at the cost of a LOT of heat; they absorb/deflect a bunch of it.
The reason they Kettle Pizza and some other attachments work is because they don't involve stuff that absorbs heat, AND they divert it where it needs to go. That said, try the other suggestions here before assuming you need an expensive accessory.
I like that idea of preheating the stone as well. Currently, it's a loooong walk for me from kitchen to kettles, so that would either wipe out the effect or burn my hands carrying that distance. But I have a gasser located about half the distance that might work for this function, AND heat up faster than the kitchen oven.
When you guys add that second chimney of unlit, how long does it take for them to light and contribute? I always get nervous when considering doing that, worrying if they won't snuff out the already-lit coals.