There've been other threads chasing this up - one of them ended up leaving me pretty confident that the Plague/Black Death idea is Victorian historical romanticism, but I couldn't find that one.
However by now, whatever the origins, it has fairly well taken on those connotations from its use in protests against mass murder weapons research and so forth. Maybe that's more important than the strict facts about the origins of the rhyme. Another aspect of the folk process. If we need it, we make it.