No, they didn't. You can't "send away" thermal energy via radio waves. At the temperatures we're talking about, thermal energy is in the infrared. That's blackbody radiation.
Nobody describes a satellite by specifying the amount of heat that it produces, but by the amount of electrical energy that it consumes.
In a communication satellite, a large fraction of the consumed electrical energy goes into the radio transmitter. Radio transmitters are very efficient and most of the consumed power is emitted as radio waves and only a very small part is converted into heat, which must be handled by the cooling system.
So in any communication satellite, a significant fraction of the consumed energy does not become heat.
Your answer makes it seem like you too missed the point. If a Starlink sends a 1000W signal to Earth, that is 1000W of power that does not heat the satellite.
The creation of that 1000W requires additional hardware and power, and only a part will be available for the transmission. The rest will be additional heat that has to be gotten rid of.
