Okay, will have another think.
In the meantime, folks with books - where does Legolas fall in? He himself sings two songs (about Boromir and Nimrodel). I think the only time he is the
subject of a song/poem is in Galadriel's prophecy via Gandalf. Is that correct?
So what's in common among all these songs, and specifically these lines? Huey, if you were a mathematician rather than a chemist, I might have looked up the number of each line in the poem to see if they make up pi or e or another famous mathematical sequence or number.
Running through authors, singers, and listeners wasn't productive, and it seems too obvious given Huey's previous responses. Throwing out ideas in a different directions, most of these songs have some importance of location. Legolas is to leave Middle-earth, Galadriel is stuck on Middle-earth, Treebeard remembers lands in Beleriand, which are also mentioned in the Luthien song, and Theoden is leaving Dunharrow for Mundburg. The hobbit song and Hey dol are not clear to me; location is more tangential, the song calls for going somewhere forward.