I really think you are right, but there's a contrarian part of me now imagining something that is almost identical with that Ballad of Bilbo Baggins video that was allegedly forced on Leonard Nimoy by terrible TV producers, magically turned good through the Beatles touch.
Who knows, as unlikely as it certainly is, perhaps knowledge of the Nimoy TV sequence was what made Tolkien protect The Beatles from embarrassing themselves on middle earth.
It wasn't forced, he did it as a favor to Charles Randolph Grean, who had produced his record albums and was responsible for that skit on the "Malibu U" variety show.
Who knows, as unlikely as it certainly is, perhaps knowledge of the Nimoy TV sequence was what made Tolkien protect The Beatles from embarrassing themselves on middle earth.