One question that pops up in my mind : could the "quality" of the 6V supply affect the quality of the sound by modifying the power of heating element (and thus the electron flow) ?
In some primitive (or high-power) tubes, the heater is also the cathode. But for almost all tubes used for audio, the cathode is a metal tube/oval with the heater filament wires INSIDE for indirect heating. So the actual current through the heater isn't really "in the circuit". Most tube heaters were operated directly from mains-frequency (50/60Hz) AC power. The thermal mass of the heater completely swamps out any effect from the sine-wave power of the heater current waveform. In exactly the same way that nobody can actually perceive "flicker" from an incandescent lamp operating at mains frequency.
Of course if circuit design or execution is sloppy, (or component failure) it is possible for the heater AC "hum" to get into the signal path. But that is an exceptional condition and not commonly found in properly designed circuits.
"Golden ears" audiophools imagine they can "hear" the difference between AC and DC power in an indirectly-heated filament tube. But they imagine lots of other things also, so that comes as no particular surprise.