The electrons are not moved by heat; they are moved by the electric potential between the electrodes. The heat goes into maintaining the cathode at a certain temperature, at which the electrons "boil off" in sufficient numbers, but that requires only a smidgeon of the energy. Though depending on it, tubes are incredibly wasteful of heat.
So you say silicon and other parts don't require heat? Check that data sheet again! If the part's operating range is from -40°C to 70°C, then that part requires heat. If the ambient temperature happens to be -100°C, you have to heat it, perhaps by installing a heating filament into the chassis.
E.g. check the TI datasheet for the NE5532 op-amps:
Free air operating temperature:
NE5532, NE5532A: 0 to 70°C.
SA5532, SA5532A: –40 to 85°C.
(That standard part is quite delicate there, only down to zero!)
Conversely, thermionic tubes for a hot environment could be built without heating filaments. If the ambient temperature is 500°C, and that happens to be quite enough for that tube's cathode, then it Just Works, like your NE5532 IC at 3°C.