The above describes standard downmixing of three channels to two. But this is different from what you've been proposing, where you start off with three channels and end up with three channels, except with the contents redistributed.

It is not different. Two channels of speakers is just a special case of three with the level of the third set to zero. Three input channels available for remixing is independent of whether some of those channels are derived from others, or not.

You appear to distinguish between special cases and the generic case, calling them different things. If we prefer, we can call what I propose a matrix transform of a vector function of the input channels. I prefer the term "remixing", to that actually.

I don't understand what you mean by the above. Are you saying that imaging is shot when using a centre speaker for a listener in the sweet spot? And how does anchoring the dialogue to the centre channel end up collapsing the front soundstage?

Yes. Add a center channel speaker when you have dialog slightly left or right of centre (which can be properly imaged with two front speakers for a listner in the sweet spot), and the dialog moves more toward the center.

This helps off-axis listeners, but hurts the listner in the sweet spot. Therefore, it should be tunable to the degree one has off-axis listeners (i.e. how far off-axis). I suppose a single center channel with a fixed level (relative to the others) suffices for most theatres, but I have never found one necessary when listening/watching alone. I do expect however, that a two channel downmix of discrete LCR sources might be worse than a stereo recording as far as placing dialog: when not downmixing 5.1 to stereo, I hear very little left and right channel dialog. With stereo recordings, I hear speakers (as in people speaking, not transducers) located quite precisely when seated at the sweet spot.

In the extreme, I've seen studies that suggest anything more than two front speakers is hogwash, and I have recordings with back to front and side to front and back effects that bear this out. But, they only work in just the right room, at the sweet spot, suggesting a lone viewer/listener with time and money for precise speakers and room treatment. Google for "phase accurate speakers" for an introduction to this kind of "voodoo" (as most discount it). The effect, however, while perhaps not as pronounced or important as proponents might suggest, is very real.