I'm not aware of the Byrds doing a song based on Psalm 23. If they did, please refresh my memory. (Memory is the second thing to fail with old age. I can't remember what the first is.)
Turn! Turn! Turn!(To everything there is a season) is based on Ecclesiastes 3. Pete Seeger, communist folk singer, put these words of the Bible to music, and the Byrds covered it and had a huge hit with it.
Even the Devil can quote Scripture.
You are correct. The song was based on ecclesiastes. It reads like logic set to poetry.