If it's dissapating through black body radiation the temperature of space isn't really that relevant to cooling.
You can radiate heat from the surface of the earth into space via the sky, but you don't lose much energy that way.
You can radiate heat from the surface of the earth into space via the sky, but you don't lose much energy that way.
That's not quite true. Black body radiation is the only reason why earth does not heat up much more. It's the only factor with a negative radiative forcing.
Solar radiation is the only main incoming source of energy, and black body radiation the only outcome with almost the same magnitude (which is huuuuge btw)
The actual surface of the earth has the advantage of being able to use the whole surface of the earth.
On earth you can use things like fans and evaporative cooling. It's far superior to making huge space radiators, but nothing is free.