I can't say in which direction the derivation went, but more than one relative from earlier generations made remarks about particular playmates with whom they "took turns sliding down the cellar door."

In each case, the following remarks made it clear that the cellar door wasn't the real attraction. It was "pickin' out the splinters" that was the real fun. (The two activities were always connected.)

All of those I remember who made remarks of the kind would have been a bit past "childhood" ca. 1930 or before, so I'd suspect that the "euphemism(?)" originated before then and appears to have fallen out of general use in the community by about then; but none of them mentioned the song so I can't say whether they picked up the terminology from an older version of the song or whether perhaps the term was just "folk-familiar" enough to have been inspiration for a song that came later than their childhood years.

It was always my impression that one wouldn't/shouldn't refer to "slidin' down the cellar door" with someone for whom there was not "considerable affection," but none of my sources were sufficiently into etymology to offer an intellectually satisfying description of precise meanings, derivations, or usage,

John