I don't know anything about the origins of the song, but it was apparently popular enough to be a common term of reference around 1910 - 1920. At a family reunion long ago there was a mention of a certain young lady during which my dad said "I used to play slide down the cellar door with ***" just so we could take turns picking the splinters out of each other's butts. Two uncles said "yep, so did I."

They all graduated from H.S. around 1920 - 1924, so I would expect the "game" was a memory from 10 or so years earlier when they were younger - but one never knowns all the truth about their elders.

John