"Perhaps by the 1970s building up Beau's character seemed essential for some reason (and Robert Preston's casting enforced it). The source material is more of a lark and IMHO got more treacly and less anarchic with every succeeding version."
Thanks for the analysis, Amiens, and I certainly agree that Beau was pretty much a plot device in the original. Of course it's natural that the stage musical would beef him up a bit and give him some romantic/sentimental significance, as it is a musical. And as for adding that song for the character to the movie, I assume that may have been partly to make the role more attractive for Preston, partly to have a new song that would presumably be eligible for an Oscar nomination, and also just to have a new song that would be sung by someone who could actually sing very well.
I do think that SOME of the changes made when AUNTIE MAME was adapted as a musical were for the better -- for example, the combination of the characters of Nora and Gooch, and the complete elimination of the incredibly tiresome character Brian O'Banion.