radiators can be made as long as desirable within the shade of the solar panels, hence the designer can pracitically set arbitrarily low temperatures above the background temperature of the universe.
Yes, you can overcome this with enough radiator area. Which costs money, and adds weight and space, which costs more money.
Nobody is saying the idea of data centers in space is impossible. It's obviously very possible. But it doesn't make even the slightest bit of economic sense. Everything gets way, way harder and there's no upside.
I don't think dissipating heat would be an issue at all. The cost of launch I think is the main bottleneck, but cooling would just be a small overhead on the cost of energy. Not a fundamental problem.
Either that or your talking out of your ass.
FYI a single modern rack consumes twice the energy of the entire ISS, in a much much much much smaller package and you'll need thousands of them. You'd need 500-1000 sqm of radiator per rack and that alone would weight several tonnes...
You'll also have to actively cool down your gigantic solar panel array