Arden Cogar Jr. has written that the grinds don't really matter, they simply feel different in the wood and it comes down to personal preference of the competitor. That is an idea at the top of skill, sharpness, and geometry.
I suppose sharpness comes down to your own definition of it, but there's no doubt it is important. You can see how an unsharp axe cuts in the wood, tearing fibers rather than making a clean cut, if it cuts at all.
The first axe here is a good example of how an unsharp axe chops (not my video, but it is an Arvika comp axe, which are known to have something of a hardwood working grind on them, so it is grind, but a dull axe performs the same):
[video=youtube;spHqaokiL04]https://www.youtube.com/watch?v=spHqaokiL04[/video]
Take an unsharp axe and compare it to a swamping grind and it will perform much the same. Or compare how an axe with a worn toe cuts at each corner of the tree (the toe is already at a disadvantage, but still).
And again, if you're cutting between 30-45* sharpness is a huge factor in safety. If you're cutting at a steep angle, 45-60*, it's safer, but also severely limits cutting and chipping.
Further, it is largely skill and placement of cuts which prevents sticking. So given all this I don't see how anyone can argue against sharpness. Comes down to definitions, but it seems the argument is being made based on a comparison of a sharp axe with poor geometry vs. a nearly sharp axe with excellent geometry. The fair comparison would be a dull axe with excellent geometry vs. the sharp axe with poor geometry, or the nearly sharp axe with excellent geometry vs. the very sharp axe with good to very good geometry. And they would function pretty much the same.
Personally, I think it is nonsense to separate the two. It is like saying one can shave with their knife but it's 1/4" thick with a 30* angle. That 'knife' won't cut wood for shit. Cutting ability is the right balance of sharpness and angle for the material you want to cut.