I agree with you about directivity. I didn't mention it because while most high-efficiency speakers are (directional) horns, not all are.

About shorting rings, I'm not sure how they could reduce odd-order harmonics. I think that mechanical suspension movements and reduction of excursion reduce third-order though, so some of the better high-efficiency drivers probably have reduced third-order from improvements in that regard, in addition to reduced even-order distorton from the use of flux control rings.

Wayne