I think it's a mistake to conflate poetry and song. Poetry is a genre; song -- melody+lyrics -- is a genre. Song lyrics are always going to qualify as verse because of their graphic form, just as many advertisements do. Any sequence of language whatsoever (consider "concrete poetry") can be arranged graphically on a page as verse, and then people who see it or who hear it read aloud have the opportunity to argue about whether it qualifies as poetry, and if so, where it lies on the continuum from Abominable Doggerel to Great Poem. There are always going to be song lyrics for which the consensus judgment is that they are poems as well as song lyrics, but that's an accident of the form. [There are of course songs that consist of already-recognized poems for which melodies have been written after the fact; for sanity's sake, I'd like to exclude them from the discussion. They're hybrids.] But a song is a-melody-plus-its-lyrics, and the power is in the music rather than in the words.
One of the perennial mysteries, and the central question of the fascinating science of psychoacoustics, is how the power of music -- a subset of which is the power of song -- is to be explained. Nobody really knows. It's a mystery that people will happily listen to the same song played thousands of times, day after day, year after year, enjoying it every single time, never getting tired of hearing it. But if you take away the melody and just read the lyrics aloud, that changes drastically; nobody wants to hear the words to a song read thousands of times aloud, not even when the popular consensus is that those words are so well written that they should be called a poem.
A lot of the comments you've posted have contained some variant of "What people like is poetry set to music"; I don't think so. What people like is songs; when they like a particular example of melody+lyric, they don't care whether that lyric is a poem or not.
For a poem to be more than just verse -- more than just non-prose -- it has to be powerful enough to stand on its own, without being set to music.