I don't think the C6th will ever become the "dominant" neck until someone gets some modern sounds from it.
But you CAN get modern sounds out of it, just not modern country (unless you include the stuff that's really just rock with country lyrics)...as Jerry notes, rock and blues work great, but I disagree with him about chord work. If you avoid that 6th, or play 3 frets up for blues, depending on your copedent you can play plenty cliche...or creative...rock chord riffs.
I play a B6, 8 string, 9+2 copedent, with the 6th on the 3rd and 7th strings (a slight reversal of sequence from standard C6) and it works perfectly well for more traditional country using E9-style A&B pedals, as well as rock oriented stuff with a 2nd and 6th string full-step pull for blues turnarounds and other "normal" rock stuff. The same things could easily be adapted to the standard C6 tuning, but not without changing the copedent away from the western-swing or Hawaiian oriented changes.
But even with the standard changes C6 is better suited for single-string leads when playing outside traditional country, which is I think what Paul was referring to. I recall reading that with Dire Straits much of his bluesier-rock work was on the C6 neck, and if you were to try to play prog rock (Yes, Gentle Giant, King Crimson) or jazzier rock (think Steely Dan) C6 seems much better suited than E9.
I'm aware that some guys can play the heck out of rock on an E9 neck - but it's not as natural, and certainly not as easy unless you add to the semi-standard 3+4 "starting" pedals/levers.
Once I found the rock basics in the B6 copedent that also included E9-type changes (essentially an early universal, but think of it in "reverse") I quit trying to play E9 permanently, as even if I play the occasional country tunes I'm all set with my B6 copedent (thanks to Sneaky, who developed it but used it in a different way).