This thread is being confused by use of an undefined term "traditional". Let's stick to "folk". That is music and song handed down and modified by the oral tradition (which precludes acknowledgment of authorship). Apply that and everything gets a lot clearer. The fact that traditional-sounding stuff is not folk does not make it worse (or better). It merely means it is not folk.

Another approach might be to define "traditional".

Interesting things definitions.

If you assert that the need for the oral tradition precludes songs first published in written form from ever being folk song the "the Cutty (or Cuddy - but not Ruddy) Wren" (1342, I think) is not a folk song. But it must surely be traditional.

I am told that "Darcy Farrow" is modern. But many think it sounds like American traditional.