I have no doubt but that it has some effect. I'm willing to accept that there's some clarinet/ mouthpiece/reed /player combination where that effect has actually caused an issue. But by and large, based upon many clarinets being pulled and played with and without rings over the years, the difference has never been apparent to me as an issue to be concerned with.
I always used tuning rings, if at all, as an indexing tool so that I'd pull to exactly the same amount on a given horn where the barrel, to suit my own mouthpiece/ reed choice, needed a mm or two pull for optimum results. This is especially useful in the event that the pull is taken between the upper and lower joints where a ring not only helps preclude "wobbling" between the joints but also helps prevent the adjustment from changing during spirited play. A 5 MM pull anywhere might well snarl things up a bit with the larger gap were one to ever do such a thing- at least for a couple of notes.
As with the effect of resonators protruding into the sax chimneys I'm perfectly prepared to believe that there exists an effect (for good or for ill or simply for "different") but also believe it to be simply insignificant to undetectable for me, for most players, and for virtually any listener.
"Virtually any" is, of course not to be confused with "every".