Different problem.
Cooling can be done in space, it's a normal problem; not easy, nothing in space is easy, but on the scale of space it is normal.
However, at the temperature you would run a chip at, a radiator you make for space will just work better, reject more heat, if you leave it on the ground.
We get convective cooling down here as well as radiative. At operational temperature, convection more than makes up for the atmosphere being a heat bath at ~20 C or whatever.
But does it make sense to do it?
No.
You are overstating the engineering difficulty. Starlink constellation already dissipates around 100MW.
(the sarcasm is intended; there are some usecases for "space servers", around stealth for sure; to "scale it like a terrestrial DC" though ... dunno; not matter how I turn it, it just seems crazy expensive)
Will starlink build its own edge ? Maybe. We'll see. Whether it makes sense to have this infrastructure "colocated in space" if you're not also the owner of the "connection infrastructure in space" though? It'll come down to cost. If I were anyone else but SpaceX ... that's where my eyes would be.