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.
In space or vacuum radiation is the best way to dissipate heat, since it's the only way.
I believe the reason the common person assumes thermal radiation is a very poor way of shedding heat is because of 2 factoids commonly known:
1. People think they know how a vacuum flask / dewar works.
2. People understand that in earthly conditions (inside a building, or under our atmosphere) thermal radiation is insignificant compared to conduction and convection.
But they don't take into account that:
1) Vacuum flasks / dewars use a vacuum for thermal insulation. Yes and they mirror the glass (emissivity nearer to ~0) precisely because thermal radiation would occur otherwise. They try their best to eliminate thermal radiation, a system optimized to eliminate thermal radiation is not a great example of how to effectively use thermal radiation to conduct heat. The thermal radiation panels would be optimized for emissivity 1, the opposite of whats inside the vacuum flask.
2) In a building or under an atmosphere a room temperature object is in fact shedding heat very quickly by thermal radiation, but so are the walls and other room temperature objects around you, they are reheating you with their thermal radiation. The net effect is small, in these earthly conditions, but in a satellite the temperature of the environment faced by the radiating surfaces is 4K, not a temperature similar to the object you are trying to keep cool.
People take the small net effect of thermal radiation in rooms etc, and the slow heat conduction through a vacuum flasks walls as representative for thermal radiation panels facing cold empty space, which is the mistake.