Yabia – referencing Hashem isn’t always in the context of shiros and praises. Many times it’s a statement, like saying “Hashem runs the world”. Also, when saying zemiros, many don’t say any of Hashem’s names because al pi halacha you have to have kavanah when you say it.
Was this teshuva trying to justify what chilonim do in Israel, where elokim just is used instead of “God”? Sounds like it was. It doesn’t hold water.