... That cap is for smoothing the dcv stand off/elevation voltage. It should be there. If the dc stand off voltage is not true dc, smooth, it defeats the purpose. It can modulate the heater causing noise.
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My understanding differs to the above.
Some valves have imperfect heater to cathode insulation, such that a current path exists which allows the AC heater voltage to appear at the cathode, causing a buzzy hum. Fully bypassing the cathode usually gets rid of that, but sometimes is insufficient, and if the design doesn't allow full bypass, then other mitigation must be used.
The heater to cathode current path is easily saturated by a few micro amps of DC, eg by biasing AKA elevating the heater circuit such that its VDC is a few volts higher than the cathode voltage. Once saturated, no modulation of the current path is possible, because it's saturated.
So pure DC is no better than ripply DC.
By the same token, the circuit common (ground) point used can make no difference.
Provided any ripple etc on the DC is not so great that it brings the instantaneous heater + elevation voltage down too close to the cathode voltage, saturation of the h-k current path is maintained.
Rather than smoothing, the purpose of the cap is to decouple the elevation voltage, ie make it low impedance (eg a high impedance heater circuit could facilitate an unintended feedback path between circuit stages), and to ensure that the heaters are balanced and referenced to 0V AC.