That doesn't sound right to me. If there's no air, then only black body radiation can be used to cool the data center. That means a massive radiator, a lot larger than a heat-to-air radiator+fan used on earth.
But geothermal cooling would be great on the moon too. Run a pipe 2 meters under the lunar surface and it is -21C.
I think the whole idea though is to make a low wattage space-stead so you can store copies of Moana out of reach of Disney cease and desist letters.
It won't stay -21C for very long, if you pump heat into it.
Really, radiative cooling is your only longer term option.
Btw, you can make your radiative cooling a lot more efficient than you gave in your example, if you run it at a higher temperature. Radiated power grows with the fourth power of (absolute) temperature. So, run your chips at something closer to eg 100C and you radiate more than 2.5x as much power.
You are right, and also it would require digging which is a lot harder than laying out panels on the surface. Back of the napkin it's a tossup depending on the conductivity of lunar sub-surface material and how much pipe you lay. Just like on Earth.
> Btw, you can make your radiative cooling a lot more efficient than you gave in your example
This is true too, heat pumps could even get higher radiator temps than 100C if you like.