First, there’s conduction and radiation as means of moving energy. (Heat is just energy). Convection - wind circulation driven by density differences in gravity - is irrelevant. Besides, currents cool things by conduction.
In a vaccuum, radiation is the only way to shed heat. Conduction can direct the heat to the exterior, and how well that works depends on the heat conduction properties. It sounds like the satellite makers have at least figured that out. An alternative would be, like an auto, to circulate coolant - but then we have a pump (moving part) and risk of coolant leak, etc. Makes the compuer more prone to fail.
However, any satellite is exposed to the energy radiating from the sun - which will heat the satellite. There are assorted tricks to mitigate this - a heat shield, held off the satellite itself, like an umbrella, minimal connections. But that too will soon heat up and radiate heat towards the rest of the satellite so it is not completely effective. The shield will include solar panels (to give power, of course) that will absorb some of that energy, but then that energy will be used in the computation power of the satellite and converted back to heat.
So the trick is to conduct the heat from the computer units to fins that radiate it into space, and then keep these shielded by the shield/solar panels. Think of it like a “T” - the flat top faces the sun, generating power, and the stem is a radiator always in shadow. That is how the ISS does it. However, this adds the attitude adjustment units - thrusters or gyros - to keep the unit properly pointed. Thrusters eventually run out of fuel, gyros (moving parts) could eventually wear out.
There are second-order issues. First, the satellite, unless it is quite a way off, will be eclipsed by the earth each orbit, so almost half the orbit (depending on altitude and inclination) will be in darkness. That’s good for cooling, bonus, but means the data center will need batteries, which means extra weight. Second, while passing in front of the earth, there will be reflected sunshine energy hitting the cooling fin(s). It’s not as intense as the direct sunlight, but still needs to be considered. The whole will be going through this constant heating and cooling cycle.
Finally, satellites need the ability to de-orbit should they fail to avoid space junk; or altenatively, some just are designed for the orbit to decay so when their useful life is done, they reenter.