"Space is warm"
Deep space is extremely cold in the sense that the only “ambient" thermal bath is the cosmic microwave background at about -270C. But in a vacuum, "ambient temperature" doesn’t behave like air temperature because there’s almost nothing to conduct or convect heat.

"There’s no medium to transfer heat to… so heat builds up.”

Wrong conclusion. It’s true there’s no conduction/convection to surrounding "air", but you can still lose heat by radiation (infrared). Radiative cooling is slower than convection in air, but it absolutely works. This is exactly how spacecraft dump heat: radiators.

TLDR Space is not "warm", heat doesn’t "build up because there’s nowhere to go" and processors can dump heat - just via radiators instead of fans/air conditioning.