If you want to redefine the phrase "traditional music", please go do it somewhere else. Find a different phrase to denote music that came from the people. Stop trying to steal the name of the genre of music that a lot of us play.

You've got a name for your precious genre - and that name is F*lk. Again I cite the objectives of The International Council for Traditional Music (formerly The International Folk Music Council, who gave us Karpeles' 1954 Definition) which are: to further the study, practice, documentation, preservation and dissemination of traditional music, including folk, popular, classical and urban music, and dance of all countries.

Surely it's the singing of the song that is traditional, (or 'a tradition') not the song itself?

The very act of singing and playing music is traditional - an unbroken continuity of musical action going back 50,000 years along with the rest of human culture. Listen to any music - any human being singing any song - and what you are hearing is the product of thousands of years of human musical creative tradition. No music is more traditional than any other.