this is already done. about a quarter of the big panels on the International Space Station are radiator panels.

they're always aligned 90° away from the solar panels so the sunlight hits them edge-on and their faces are pointed towards empty space.

a satellite with data-center level servers on board would need an absurd amount of both solar panels and radiator panels, so many panels that it would need to orbit much higher than the ISS, because of trace atmospheric drag.

Satellites still have cooling (radiators, heatpipes), but with only radiation they are orders of magnitude less efficient. So having a satelite that is like 0.1% datacentre & the rest just for cooling is kinda a stupid idea (and that is before adding solar panels, that themselves too have to be cooled the same inefficient way!).

It's not impossible or anything, just really stupid, especially if the point is easier cooling. It's not surprising knowing the source tho (dude prob just saw undersea datacentres & thought space is cooler).

I'm not sure if the term radiator is what's making this sound appealing, but on earth radiators mostly use convective and conductive heat transfer to transfer heat off the radiator. The energy in the radiator fluid is transferred to the radiator tube shell and then to the air.

Temperature is a measure of how much kinetic energy the molecules in a system have. The absolute coldest place possible is a total vacuum, because there are no molecules. But if there are no molecules, you can't have convective or conductive heat transfer.

If you put radiators in space you'd get conductive transfer from the radiator fluid to the shell, but there are so few molecules in space to have the energy (heat) transferred to that there is almost no conduction or convective transfer off the shell.

As the person the the OP mentioned, there is still radiative heat transfer where the kinetic energy of the molecules convert to electromagnetic energy. The rate is described by Stefan–Boltzmann law which uses this tiny ass constant and the temperature of the radiating body to the 4th power to describe the rate: σ = 5.670374419...×10−8 W⋅m−2⋅K−4

So, plainly put: it's slow as fuck.