I think you're running together two or three different questions. Can we identify any songs which have irrefutably been passed down from voice to voice since the year dot, without ever being written out and tidied up? No, of course not. Can we identify some songs as being more 'written' than others? Yes, clearly - some 'traditional' songs seem to have originated as broadsides, others were essentially music hall numbers, and quite a number seem to have been written by a certain B. Dylan. I'm quite happy to sing a song by Bert Lloyd, but I'd rather not announce it as something it's not. The charge against Lloyd isn't that he wrote songs but that he denied doing it.

If you're asking whether any of this matters, the answer is that it matters to those it matters to. At the end of the day the songs are out there, and long may they be sung.