STARTING ON JAN. 23, when Pete Buttigieg announced his exploratory committee for a presidential run, the mayor of South Bend, Ind., has largely blazed the campaign trail without a blazer. Mr. Buttigieg, 37, the first openly gay major-party presidential candidate, has become the surprise upstart of the 2020 Democratic race, and a surprisingly casual one, too--appearing at nearly every campaign stop and media opportunity sans sport coat, letting his pressed white shirt (its sleeves always rolled) suffice and his usually-blue necktie fly free. In the past, candidates from Barack Obama to George W. Bush occasionally went...