I am a great believer in the assertion that people who are singing ordinarily know what they are saying. (The exceptions are few and notable.)
A pinsman, in the OED or not, I think we may take as a seller or maker of pins. Pins used to be very scarce before they were mass-produced, and itinerant salesman of these and other simple goods (needles and thread, for instance) roved the rural landscape, welcomed by housewives who needed such things. Little girls (and boys?) also sold cards of pins on the streets in cities.
Needles and pins, needles and pins ...
I think "pinsman" is literally correct.