Current into a low-resistance "heater" element is used to produce the heat required for Thermionic Emission [0] in a vacuum tube. You only need the heater/emitter to be hot, and insulating the tubes would just spread the heat around to everything inside of it — at some extreme, making everything into an emitter, instead of elements that control the emission.
This is not really accurate. To get meaningful emissions from normal electrodes, you need to heat them up to about 2000 °C. Vacuum tubes operate at 700 °C or something like that. The trick is that one electrode is doped with special rare-earth additives that greatly increase electron emissions. The same treatment isn't applied to the rest of the device. So, even if all internal components have the same temperature, a vacuum tube can still work (to some extent).
100% correct and I appreciate the additional details! I couldn't come up with a good analogy to explain you want the emitter as a separate and unique element from everything else involved in a tube — oversimplified in the process.
[0] https://en.wikipedia.org/wiki/Thermionic_emission