I might be crazy but does anyone see an issue with trying this? Please stop me before I buy a new rectangular stone.
I see a few issues with your concept.
First, a pizza stone is not intended for uneven heat distribution and fast heating - used in an oven or barbecue, it needs time to heat and this ideally happens slowly and with a distributed heat source. Yes, chamotte is more resistant to thermal shock than real stone or clay, but it has its limits. There is a reason that while it’s possible to wipe ash from a wood-fired oven with a wet rag (if you know how to and do it swiftly), many oven manufacturers will explicitly warn against doing so.
Second, the surface is rough, it will never be as smooth as metal or glass-ceramic. This can abrade and damage cookware, especially if it’s shuffled around as a French top is designed to.
Third, the porous surface makes cleanup a nightmare, particularly when you also remember the roughness of the material.
In short, I don’t think it’s a viable and practical idea.
If you're planning to buy a new rock anyway, consider buying a soapstone griddle, which are intended for use on the stovetop (and can also be used in the oven, if you like.)
Though it appears from a quick look that buying a steel plate would be a closer replication of a "French top" and probably cheaper.