Again, no air...

Yes space is cool but special care needs to be taken to cool things. Only radiative cooling works, so special structures like antenna have to be built to "radiate" heat into space.

Otherwise space structures will overheat.

Not at all. All conventional cooling systems on earth work by blowing air on a radiator. You need to transfer heat somewhere else. Radiators, despite the name, rely on conduction and convection for 80-90% of their cooling capacity. In space, they can't conduct or convent, only radiate heat.

The ISS has giant radiators larger than the main station to cool it down and all those combined account for 70kw of cooling. That is 35 J19 pros worth of cooling. A nearby bitcoin miner to me is 500 MW. Many data centers are similar in scale. That's 7,000 times the energy production and heat dissipation requirement. It's just incredibly inefficient to use and dissipate a lot of power in space.