From googling around, most rock melts at around 1000C, so I guess the biggest challenge is making a reactor that will survive heating up to 1000C? The melting point of uranium is barely 1100C, so seems like an awful thin margin.

How would they handle not irradiating (and generating all sorts of isotopes in) the tunnel walls? A 10MW reactor generates a fuckton of radiation.

Nobody could come anywhere near the "melting" head, even after shutdown.

Wouldn't hitting even a tiny amount of water would be catastrophically bad, causing a steam explosion? Hitting oil deposits, coal...minerals that contain water...