Yes, most melodies have implied harmonic structure. What is more true, perhaps, is that many traditional tunes don't have a chord pattern. The accompanist does have to follow the structure of the tune. That being said, lots of tunes do have patterns. There's about 100 Em reels that all have the same chords, all in the same places.

Trying to make rules about what is or isn't included in the performance of traditional music is a very difficult undertaking. I don't think anyone will ever be successful at it, since there is no consensus amongst the traditional music practitioners. Perhaps the only rule should be "Don't tell others they are doing it wrong."