So what's silly about the white key explanation? It's true isn't it?
There isn't much point in knowing something just because it's true. You want it to be interesting or useful.

"Click Go the Shears" is a Kiwi song, to the same extent that the Skye Boat Song is Scottish and The Star Spangled Banner is American. Obviously the tune came from somewhere else. Maybe the words did too - the usual versions seem to be Australian - but nobody knows exactly where or when it was written (maybe as a parody of a Temperance song). It doesn't matter a lot since the shearing culture in NZ and Australia was much the same and the song spoke to people both sides of the Tasman.