It's either one, zero, or infinite. Pretty sure it is one of those three.
I will run some simulations and get back to all y'all on this.
This may be a joke, but I see a lot of truth to it. If we are to approximate the dispersion of most sound sources as omni (even though there's evidence this is wrong), and if we want to re-create the experience of having those sound sources in front of us authentically, we need to have an omni speaker located where each source was located (relative to the listener). Having two separated omni speakers create a phantom center is not going to generate the same room interactions that a source actually in the center would.
I think I've read and considered the implications of each post in this thread (but not all links, etc). I'm fairly well convinced that a center channel speaker should have dispersion that's equal in the horizontal plane from +90 to - 90 degrees, and have no dispersion behind it, since any reflections from behind it should already be in the mix (either recorded authentically, or placed there by the sound designer). This thread has had little attention paid to vertical dispersion, and whether it should be broad or limited or if it really matters, and if so how. I have thoughts on that from prior experience and research. I'm not sure what dispersion stereo speakers should have ideally, even though that's the primary focus. Based on this thread, I'm also not sure whether when I listen to that center channel, I should have my walls treated or not. I suppose a speaker designer need not concern himself with such questions as wall treatments.
I will suggest the following thought experiment (to all thread participants):
Consider going to your favorite venue, and recording your favorite live performance with a single omni-directional microphone, placed in your preferred location. All the reflections of the venue will be captured with proper levels and delays.
Now consider playing back that recording on a single speaker (your dream speaker), and listening to it in an anechoic chamber.
How do you think you would like the sound?