Pure speculation, but perhaps the reason why he wanted the song cut was because he felt the scene was narratively unnecessary. For the sake of argument, I might even say that I agree with him: the scene doesn’t advance the plot. However, plot is only one dimension of a movie. The “Moon River” scene is essential to the film—imagine Breakfast at Tiffany’s with the scene omitted. What is lost? Holly singing “Moon River” contributes nothing to the plot, but has everything to do with mood or atmosphere. The song’s subject—wanderlust, the lure of the river—deepens our understanding of Holly’s character and also invokes a wistful, melancholic mood that haunts the rest of the movie. The scene is an instance of diegetic music, music that is actually played within the fictional space of the film’s story.
Of course, the movies are loaded with such moments, and the best of these moments remain indelibly etched in our memories. For author and critic Umberto Eco (1985’s “’Casablanca’: Cult Movies and Intertextual Collage”), the greatest movies are “great” precisely because they contain such moments. The challenge presented to us is how to write about such moments in a way to do them justice. As Christian Metz stated in his now famous epigraph (cited by James Monaco in How to Read a Film), “[M]ovies are difficult to explain because they are easy to understand.”