What you call the egg song is in fact a call for a rooster. It's got little to do with laying eggs as you have found out.
https://www.backyardchickens.com/articles/the-egg-song-it%E2%80%99s-not-about-the-egg-it%E2%80%99s-an-escort-call.74386/
At a quarter of a mile away your hens can hear the...