Stanron, did you note MGM Lion's quote from Dan Worrall about ten posts down the thread (he was contradicting me, as it happens!):
"The concertina was also frequently used to accompany singing. Such singing was most typically done during evening slack time, and involved both nautical songs as well as any popular songs of the day".

Walter Pardon sang and played melodeon, but not at the same time. Plenty of people who sing and play box find it confusing and difficult to do both together, but quite apart from that, our 'folk' singing tradition seems to have been predominantly unaccompanied, so perhaps it was simply the 'done thing'