Electronic – In a vacuum triode, what prevents the grid from acting as another anode

These were common before the transistor and called "Phototubes". Every 16mm sound projector in the US schools had one to read the sound track on the film.

With higher voltage and multiple plates - one designed to emit electron when struck by light. The others are meant to kick out more electrons when they are hit by an energetic electron and you get a cascade. These are "Photomultipliers" and are still used extensively for photon counting and any very low light application like fluorescence spectroscopy or phosphorescence lifetime measurement.

There is a very fast and sensitive star tracking application that uses a spinning mask in the optical path to a photomultiplier. I have not seen one in decades even though they have a much faster response than CCD based tracking.

Hamamatsu is one maker of both devices in many forms.

Possibly shouldn't be an answer, but...

The TL/12 (original, 1947) has separate bias (cathode resistor) for each valve; the others generally used a common cathode resistor and there, you do need reasonably well matched valves. You are correct that none (that I know of) used negative grid bias supply, sorry if I misled you on that.

I'm puzzled how you expect the two halves of a classic push-pull amplifier would interact. Each has its own DC feed via half the transformer primary; its own cathode resistor (if automatic bias, except as above) or its own grid bias (if not).

For DC purposes, these are practically independent circuits. Interactions between them are pretty much limited to variation in the main supply voltage, at the centre tap of the primary - which introduces second order effects.

It's not like a transistor amplifier (excluding the transformer coupled ones!) where both halves of the push-pull pair are in series in the same circuit and cannot be biased independently.

So the procedure (for classic push-pull amplifiers) is :

bias first valve to required anode current
bias second valve likewise
check that first has not shifted appreciably
done

And for the Leaks, with automatic bias, that just meant check the voltages across the cathode resistor. Whether you simply replaced valves if out-of-spec or tried to get another year by fiddling the resistor was up to you and your bank account (or valued sources of parts!)

Mismatch between the anode currents will result in modest levels of even harmonic distortion. The two DC currents in the transformer balance out so there is no reduction in available flux before saturation, unless the mismatch is gross.

I've seen people (wearing gloves!) pull a valve out and plug in another without noticably distressing the amp (Not recommended though!) I even ran a TL/12 for a couple of poverty-stricken years with an EL34 and a KT66. Sounded lovely, but looked a bit too much like Laurel & Hardy!

Now there may be some reason why the Thorens gives unusual grief, but so far I can't see it. If this is the right schematic the bits I can see look like straightforward independent automatic bias (like the TL/12), and I'd guess the grids are grounded, giving about 35mA in each anode.