Quite a lot of commonly-sung songs *did* start out on paper. You probably know Johnny Cope, Erin go Bragh, Loch Lomond, Ye Banks and Braes, Gloomy Winter or The Wild Rover?

Look at enough broadsides and it's pretty clear which were newly composed.

Or, a point that came up in the thread about "Blowin' In The Wind": how many people want to hear folk-processed versions of Bob Dylan songs? They're a lot more fixed than anything by Mozart. "The" sheet music for them is completely unproblematic, the definitive versions are Dylan's (there may be more than one but not a great range of them, and you aren't going to find one in a different mode or time signature).