True enough, Michael, there are also probably some songs where only a "careful" harmony is appropriate.

For example, in certain modes, you should never inject a third because you'd have to choose either the major third or the minor third, and the song "wants" that ambiguity to be preserved; however, it might be perfectly acceptable to sing a fifth above the melody, or to sing a "drone" on the tonic note.

For some usually-unaccompanied songs, injecting a second voice can be problematic not only because of the harmony-vs-unison issue, but also because the tempo/rhythm is purposefully, um, vague. Perhaps "unstructured" would be a better word...