To get back to the (very interesting) questions in the original post: I think it unlikely that traditional singers always used the same key for a given song if they were singing unaccompanied. This would entail having perfect pitch, which not many people have, or else having an instrument or pitch pipe to give them the starting note or key.

I know Bob Copper used to have a tuning fork to set the pitch for their family songs, but I don't know how long this has been the habit in the family. I'd guess that most singers who sang in pubs or private homes didn't use a tuning device, and just plucked a note out of the air.

It's important for a singer to find the right key for each song - but this means the right key for the singer too. Songs with a big range will leave you either screeching or growling unless you get it right. A different singer may require a different key to sing the same song comfortably. It's even more crucial for harmony groups, as the total range covered is likely to be greater, and one part can easily get squeezed out at the top or the bottom if the key isn't right.

I wouldn't be at all surprised it Sharp et al didn't transpose the songs to a key that was easier to notate or to write a piano accompaniment for. And no, I don't think it changes to song that much if you alter the key. It's the same song - Happy Birthday is always the recognisably the same no matter what key it's sung in.

If instruments are chosen to accompany the song, then that's what they should do. The singer's key preference should be respected, as voices don't come with capos (alas). If the instrument can't play in a key that suits the voice, then it's the wrong instrument for this particular singer and song combination.

Marje