You can radiate heat into the vacuum of space, it’s just extremely slow compared to doing it into atmosphere. Vacuum is not a perfect insulator in this regard.

Think of it this way, if a vacuum was a perfect insulator, how would the sun radiate heat to Earth?

Your car radiator is actually using convection to convect heat into the air.

The spacecraft radiators use radiation to dump heat by emitting infrared photons. Photons do not require a medium. This type of radiator works by maximizing the area of hot surface exposed to empty space (which has an effective temperature of 3 K). They have to be pointed into a dark area and away from the sun. There's no advantage to fins, because you want to maximize area perpendicular to the dark sky.

Both devices are called radiators, but they are different kinds of devices.

The radiators dissipate the heat as infrared radiation. They work as long as they are pointed away from the sun or earth.

If they couldn't get rid of the heat, there would be no satellites or space stations.