PETER PAN with Mary Martin was my very first Broadway show, in 1954 when I was 13. I do not think that Mary Martin's vocal during the "I'm Flying" sequence was pre-recorded.
When I saw 1958's GOLDILOCKS with Elaine Stritch and Don Ameche, Margaret Hamilton(the Wicked Witch of the West in the WIZARD OF OZ film) was also in the show. At the end of the number "Two Years in the Making", the final high note is lip synched by Miss Hamilton, while the actual high note is sung and credited by another singer, who I always thought sang it live from the wings, but I suppose it could have been pre-recorded. At any rate, Margaret Hamilton was a joy to watch.
"Madam Rose...and her daughter...Gypsy!"