When people sang unaccompanied, they usually did so when working. You can't really play an instrument when working a loom, or driving a forklift, for that matter. Not only that, but it's hard to sing parts BY ONE'S SELF. They would also sing musical hall songs UNACCOMPANIED, melody only, despite these songs usually being performed WITH accompaniment and parts.
There is a good body of evidence (such as west gallery music and pub pianos) suggesting that they could and would accompany songs when possible.

The guitar is far more versatile than the cittern, so why would MC drop an instrument which has been an extension of his being in favour of an inferior one just because it once had English in the title?

Don, superb post. I've run into a similar problem in my personal life, which has just created immense problems for me and someone else. This, in a category of immigration which Wav supposedly approves of.