Interesting question about the wizards. I tend to think that they didn't sing because they were "undercover" during their tenure in Middle-earth; clearly, it was intended that they be thought of as old Men, not as Elves or Ainur. I can't recall any instances in which Men used music as a means of magic or power, and if the wizards were to do so, the immediate presumption might be that they are some kind of strange Elven-kind. It's possible that at some point earlier in his time as one of the Istari, Gandalf at least did use singing as a means of implementing his personal power, since the Men of the North gave him the name Gandalf, which Tolkien at least once translates as "elf of the wand," or "elf with a staff." With that mistake behind him, he may have eschewed further uses of music to avoid becoming thought of as an Elf.
Though the wizards don't use music, they do use Words of Power, and as fans of
The Music Man know, singing is just sustained talking.
In some traditions, "songs" are not words set to music, as we think of them, but intricate poems. So it's possible that "music" might be used by the wizards in the form of recitation, the words providing meter and rhythm and tone, if not specific diatonic or modal pitch. I should go look at the text, but it seems to me that there are indications that some of the songs in LotR are "sung" in this way (I'm thinking of the Song of Luthien, the Song of Durin, and Bilbo's song about Earendil). If such is the case, then it would make sense that during this mission, the music of Istari was the music of carefully chosen words, spoken in particular meter and tone.
Well, it's another thought.
