Good posts Steve. The thing that puzzles me is the "turning voltage into current" explanation. Now I haven't completely thought this through...its an impromtu post, but help me think this through...
Sure.
By the way, "turning voltage into current" is basically just their way of saying "impedance transformation."
From Kirchoff's law we know that current is preserved in a closed system. So current out = current in. Pot is a voltage divider so part of the current is going to ground, part is going to amp. The primary side of a tranny has a DCR which makes me think it too acts like a voltage divider, so why would this be any different for a TVC/AVC than a pot? I can only surmise it has to do with AC properties rather than DC properties (i.e. reactance).
Well yes, the DCR of the primary as well as the secondary limit how low the output impedance can be and do effectively act as a voltage divider with the load impedance and attenuate the signal to some degree. To what degree will depend on the winding resistances, the turns ratio, and the load impedance. But with a decent design driving an appropriate load, this attenuation is kept small compared to the attenuation provided by the TVC.
Keep in mind that the DC resistance of the primary gets reflected as the square of the turns ratio just as the output impedance of the source does, whereas the signal gets attenuated simply by the turns ratio. So if your signal is attenuated by a factor of two, the DC resistance of the primary is dropped by a factor of four.
So with respect to primary and secondary DC resistances, the TVC is at its worst case at 0dB of attenuation. Where by comparison the resistive attenuator is at its best case. Though most don't often listen with the volume cranked all the way up so it's rather moot.

Anyway, getting back to the "turning voltage to current" as I said that's another way of speaking to the impedance transformation that transformers can accomplish.
The classic example of this is the output transformer on a typical tube amp. While the tubes are run from relatively very high voltages, their weak point is that generally speaking, they can't handle as much current as say a transistor can and if you try and drive a low impedance load like an 8 ohm loudspeaker, you're not going to be able to get any power to speak of.
If you can't muster enough current directly, but you can swing a lot of volts, you can use a step-down transformer to transform the low impedance of the loudspeaker to a higher impedance that the tube can handle without current limiting.
But of course there is a price. Because the transformer is a step-down, in order to get enough voltage across the secondary and subsequently the loudspeaker in order to get the power you want, you gotta swing a lot more voltage on the secondary. Which means you have to build up a lot more voltage gain than you would otherwise have to if you had the current capacity to drive the loudspeaker directly.
The one saving grace is that as I said above, impedance gets reflected as the square of the turns ratio while the voltage gets reflected by the turns ratio so your current goes down faster than the voltage requirement goes up.
Anyway, that's the gist of what they're getting at when they talk about "turning voltage to current."
se