I actually took a History of American Musical Theatre class last semester in college, and our textbook was Our Musicals, Ourselves: A Social History of the American Musical Theatre by John Bush Jones. I LOVE this book. He basically takes each decade of America 20th century and writes about what was happening in that decade and how that translated to the stage in musicals. Which sounds like exactly what you want. There's World Wars I and II, Depression, Roaring Twenties, Cold War, Black and Jewish musicals, Issue musicals in the civil rights era, etc. I highly recommend it.
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