There is no evidence that the bawdier version of John Anderson is any more original than the other one. It's quite likely that Burns made it up in its entirety.

And I suspect that any bawdy interpretation of "Auld Lang Syne" is entirely in Freeman's head. He appears to think that "folk" = "crude and stupid" (and also that crudeness and stupidity are a good thing).

There were bawdy words for the tune we now sing "Auld Lang Syne" to, but they have nothing to do with impotence.

Given some of the other uses for the older "Auld Lang Syne" tune (like an extremely longwinded and brutally insulting political satire), it can't have been seen as "melancholic" at the time. It just isn't very good compared with the alternative that won out.