Thank you. I'm glad that someone else saw her perform that song on a TV show. I was beginning to doubt my memory, but it was hard to believe that I could have such a vivid remembrance of seeing Lansbury do that number on TV if it had not actually happened. I totally forgot that my parents watched The Perry Como Show every week.
My grandmother used to watch The Eddie Fisher Show and (of course) The Ed Sullivan Show every week and The Arthur Godfrey Show every weekday morning. Those TV shows (among many others) regularly featured Broadway performers doing songs from musicals, but it's hard to find a listing of many of those performances via the internet.