This approach can make the Fisher-Yates shuffle arbitrarily accurate. It would be possible, but messy, to apply it to sort values that compared equal, too.

I fail to see the difference. Certainly one can make any numerically-limited algorithm "arbitrarily accurate" if one is willing to increase the number of bits used to represent the numbers in the calculation. The variation of Fisher-Yates that you propose would require a special check to handle the highly improbable case that the random number generator produced a number that was exactly equal to k/N, for any integer k in 0..N - 1, in order to generate more bits to break the tie (without this provision, the algorithm is identical to the standard Fisher-Yates as far as the uniformity of the sampling is concerned). Likewise, the tag-sorting shuffle algorithm I posted would need to be modified to handle the highly improbable case that two of the random tags happened to be identical (which would result in a stable-sort artifact), by generating more bits to break the tie.

...but Fisher-Yates wins by being O(N) instead of O(NlogN).

Yes, of course, but the speed superiority of Fisher-Yates has never been in question. My position all along has been limited to defending the algorithm I posted against the claim that it was logically flawed in the same way as the sort-based algorithms discussed in When the Best Solution Isn't are. The problem with those algorithms would remain even if we had infinite-precision computers at our disposal; this is not the case for the sort-based algorithm I posted. Furthermore, in comparison to the errors incurred by those truly flawed algorithms, the errors incurred by numerically-limited algorithms like Fisher-Yates or the one I posted are entirely insignificant.

Update: Fixed minor typo/error above: the range 0..N - 1 mentioned towards the middle of the first paragraph was erroneously given as 1..N - 1 in the original post. Also, immediately before in the same sentence, the reference to the variable k was missing.