WHOA THERE!! WRONG-O!!
The song is called "Distant Melody" and was written to replace the song "When I Went Home" because When I Went Home was considered too depressing (imagine that! Distant Melody is already depressing!).
Peter DOES sing the lullaby. Wendy asks Peter to sing a lullaby to his children, and Peter remembers this song, "Distant Meldoy" that the audience knows his mother sang to him before he ran away to Neverland. This song, in turn, makes the Darling children homesick, and they tell Peter they want to go back home to London.
And that is today's PETER PAN lesson.
"The Spectacle has, indeed, an emotional attraction of its own, but, of all the parts, it is the least artistic, and connected least with the art of poetry. For the power of Tragedy, we may be sure, is felt even apart from representation and actors. Besides, the production of spectacular effects depends more on the art of the stage machinist than on that of the poet."
--Aristotle
