Electronic – Why are thermionic valve/tube heaters generally designed to be 6.3 V

Some tubes use directly heated cathodes. The filament is called the 'heater' in tube land and the part that the electrons are emitted from is the cathode. (The term 'plate' is reserved for the anode where the electrons are collected). Most use indirectly heated cathodes, which allows heating with AC. I have seen a lot of heaters made with very fine filaments with an oxide coating, usually barium oxide. The oxide is fragile and it evaporates over time and leads to the burnouts or shorts. In some heaters the filaments are packed into a small space and go back and forth and heat a cylinder of thin metal with the oxide coating on the outside.

Why the oxide? Tungsten has a work function of about 4.5 eV or electron volts. The electrons must have this much energy to escape the metal. To give them the energy, the heater is run very hot, like 2700C like a light bulb. With the oxide coatings the work function is about 1.1eV. The energy of the electrons in the heater goes up with temperature exponentially. So, getting really hot makes a big difference. But so does reducing the work function and running at lower temperature which uses less power and extends the life. At 1.1eV the heater can be a dull red heat instead of light bulb white.

Heaters are not replaceable because the glass tube is sealed at both ends (or a sealed metal can for some devices) and maintains a hard vacuum. In fact, after as much air is removed with a vacuum pump, the tube is sealed and a "barium getter" is heated electrically to react with any remaining gasses. The result in a very good vacuum. A side-effect makes the mirror-like spot on the side of the tube.

A toaster needs to produce heat, like 1500 Watts, so a large wire is used and a lot of current. The tube needs to have a hot cathode which can be very low mass and is in a high vacuum. A fine tungsten wire will get hot with less current and consume less power. But it will also be easier to damage with heat and vibration.

Direct heated tungsten was used until about 1930. Tungsten with a little thorium in it was found to have a lower work function and was used until the barium oxide effect was discovered.

The RCA receiving tube manual RC-23 is quite vague about the amount of permissible variations, however it does warn that insufficient filament voltage can lead to degraded performance and reduced tube life. Of course excessive filament voltage can also shorten tube life.

Since tubes were normally operated from unregulated mains transformers, I would think your -11% would be acceptable, but perhaps not if that nominal voltage also has a large tolerance and/or you're working the tube close to the maximum ratings.