The superconducting transition temperature (Tc) of many superconducting materials increases as the pressure applied to the material is increased, so I guess the idea is to make the ambient pressure high enough that Tc is below the ambient temperature.

My instinctual reaction (as a physicist studying superconductors, but with no expert knowledge of geophysics) is that this won't work for most (possibly all) materials, because (a) the maximum Tc under high pressure is still generally less than ~ room temperature, (b) the ambient temperature increases with depth below the earth's surface.

Exactly.

Now a massive computing center in Antarctica however might have some merits.