what reason could a collector have for transposing a tune and adding a base harmony if they all they were doing was making a copy of some sheet music, as opposed to collecting it from someone's singing or playing?

Chappell wasn't a collector, he was a musicologically inclined anthologist. PMOT was intended for people to sing to at the piano - the accompaniments added to the commercial viability of the project (and Chappell would have been a good judge of that given his role in the family firm). Putting the tune in G was the best choice for voices in the soprano/tenor range (always the first one any collection was published for.

B flat wouldn't have suited as many singers. This suggests Motherwell may not have changed it from whatever the source was - some alto or bass, perhaps.

We know where Motherwell got a lot of his material. One source was Andrew Blaikie, whose manuscript I have transcribed into ABC in my website. Blaikie doesn't give his own sources. I think C.K. Sharpe was another one - if so he will have stated his source, he was far ahead of his time in scholarly standards.

Bruce Olson told me he'd been analyzing Motherwell's tunes and finally got fed up with it as everything seemed to come from some unattributed and garbled earlier source that he already knew about. He didn't mention Little Musgrave though. But if anyone's found an earlier copy of what Motherwell printed, Bruce will have done. The archive of his site here would be worth a look.