I think it's unwise to be dogmatic about "the tradition", as if it were uniform and unchanging. It may be true that most singers sang solo and unaccompanied, but there were exceptions, such as Bob Roberts who sang with melodeon. Harmony singing may not have been common, but it certainly existed - the Coppers and the Millens are probably the tip of a lost iceberg. Harmony singing and fuguing are certainly part of the Sheffield Carols tradition.

Folk singers did not exist in an isolated bubble, cut off from other influences. They were exposed to the popular music of the day, as well as religious music, which included harmonised arrangements of the type found in West Gallery music and the Sacred Harp in the US. Many folk musicians had some formal musical training, in church bands and choirs or perhaps the military, and some could read music. Also, don't forget that brass bands and concertina bands were an important element of many working-class communities. I suspect that a great many traditional singers and musicians were better musically educated than many present-day folkies. It would be surprising if some of these influences didn't affect the way some performed their more traditional songs.