Shortly before her death at age 91 in 2013, Esther Williams revisited some of her old MGM musicals. “I’m watching this beautiful young girl,” she told her son Ben Gage. “She’s so pretty and charming.

Even some huge musical theater lovers aren’t thrilled with the more movie-centric direction the Great White Way has taken as of late. (Think: musicals that are based on movies like “Mean Girls” or the ...