It makes sense to target a higher operating temperature, like 375K. At some point, the energy budget would reach an equilibrium. The Earth constantly absorbs solar energy and also dissipates the heat only by radiative cooling. But the equilibrium temperature of the Earth is still kind of cool.

I guess the trick lies in the operating temperature and the geometry of the satellites.

That's not a new problem that no one has dealt with before. The ISS for instance has its External Active Thermal Control System (EACTS).

It's not so much a matter of whether it's an unsolvable problem but more like, how expensive is it to solve this problem, what are its limitations, and does the project still makes economic sense once you factor all that in?

I agree that data centers in space is nuts.

But I think there's solutions to the waste heat issue

https://www.nasa.gov/centers-and-facilities/goddard/engineer...

Just have to size radiators correctly. Not a physics problem. Just an economic one.

Main physics problem is actually that the math works better at higher GPU temps for efficiency reasons and that might have reliability trade off.