Had to repost this brilliant comment from Brian Holton, above:
If you listen to great traditional singers … rhythms they use are subtle, sophisticated and not at all mechanical. Now, if you strum a guitar or thump a piano or bang a bodhran, you are likely to completely lose the lilting breath-rhythms of the tradition. I dare say it was happening in Burns’ time,when the piano was coming in, and perhaps O’Carolan’s Ireland wasn’t immune to that either - maybe the split between tavern and royal court was there long before the drawing-room/bothy divide, too (damn lutenists ruining the beat!).
This hits the nail on the head regarding the difference between unaccompanied singing (or playing, for that matter) and any sort of backing.