Hmm! I question the assumption behind this thread. Art songs and pop songs are written with chords, but folk songs fundamentally have only melodies, and it's up to the person providing the accompaniment (if any!) whether to use chords at all (rather than, for instance, unison, a single line harmony or a drone) and if so which chords.

That said, if a tune seems to work well with I and IV but not V, that does say something about the structure of the tune and possibly the mode. This might apply to some of those tunes that seem to imply a particular tonic but end somewhere else (just speculating, without an example in mind).

Richard