This is such an interesting theory! Some, but not all of my hens sing the egg song, but I don’t have a rooster.
Almost all (nine) of my hens hesitate before they cross our road (an unpaved easement, really). Bridgey will call at the edge of the road (a different song from the egg song) until I arrive and will accept me as an escort across the driveway/easement. Several of them will
silently (without a call) accept me as an escort. I would love to have a rooster some day to see how this behavior would shift. But I must admit, I love being their rooster. Thanks for Shari g your observations and theory!