There is a large region of the upper atmosphere called the thermosphere where there is still a little bit of air. The pressure is extremely low but the few molecules that are there are bombarded by intense radiation and thus reach pretty high temperatures, even 2000 C!

But since there are so few such molecules in any cubic meter, there isn't much energy in them. So if you put an object in such a rarefied atmosphere. It wouldn't get heated up by it despite such a gas formally having such a temperature.

The gas would be cooled down upon contact with the body and the body would be heated up by a negligible amount

These satellites will certainly be above the themosphere. The temperature of the sparse molecules in space is not relevant for cooling because there are too few of them. We're talking about radiative cooling here.