Depends on the stone. The coefficient of heat flow in stone is very low, so the heat will take forever to flow through the stone. But it should be quite evenly warmed at the top in a few hours if you keep feeding the fire. And it would be more or less immune to flare-ups and other issues.

However, you’d need to use a big slab of granite or something with more-or-less uniform properties, zero porosity (so no moisture or air bubbles are expanding or flashing to steam), and there’s still a chance that fast heating or cooling causes it to crack.

Or just use a metal slab with a pizza stone on top.