Love this thread -- so many good thoughts! It can go on forever, being so subjective (arguing who is "traditional" is rather like arguing who has a "big nose"... everyone knows what it means to them, but how do we quantify it, hmm?)
Which composed songs have made it into trad? My nominee is "She Moved Through the Fair."
Now I think I'll just stick my finger in my ear (though some may accuse me of sticking other digits in other orifices) and go sing something trad, like a nice 12th century Australian chant, or maybe Casey Jones.