Granite and similar materials are commonly cut to shape with diamond saws. This has a few disadvantages: even abrasive-grade diamonds are still somewhat expensive, the process produces carcinogenic dust of crystalline silica, and, with the usual circular-saw blades, the cuts are necessarily fairly straight and not very precise.
Granite, as I understand it, gets most of its strength from the interlocking quartz crystals that make up a significant part of its bulk. Other major constituents such as feldspar and mica are mostly much softer and less chemically inert, so they are easier to cut. (Zircon is an exception, but it typically constitutes only a small part of granite.)
Quartz can be converted to water-soluble sodium silicate waterglass with an aqueous solution of sodium hydroxide heated a bit above room temperature, ideally to 100 degrees. So a plausible approach to these problems is to cut preheated granite with a wire saw flood-lubricated with hot lye. For the wire, you’d need a metal that was adequately stable in the strongly reducing, warm, wet, and alkaline conditions we’re talking about; I think ordinary steel wire would work fine, but if not, surely some kind of brass or bronze would be adequate, in an argon chamber if all else fails. Using a wire made of a hardened tool steel would increase the risk of wire cracking but reduce the abrasive wear on the wire. Some saw-tooth-like surface treatment (perhaps a helical ridge) might be effective at increasing swarf clearance and concentrating abrasive wear on a non-structural part of the wire.
The side forces on the “sawing” wire would be much smaller than the forces on a diamond saw or even an abrasive-sawing wire, and it could cut paths through the stone with tight curves and corners, enabling the fabrication of a much wider array of shapes than a circular saw can, perhaps at higher precision, and certainly with much smaller side forces on the machine guiding it.
This approach could also work for cutting other troublesome materials that are vulnerable to lye, in particular including glasses such as soda-lime glass and borosilicate, but also some other silicate minerals and metal sulfides.