COUNCIL BLUFFS, Iowa — As Peter Johnson and Emily Neal waited for Senator Kirsten Gillibrand to arrive at Barley’s, a brick-lined sports bar in southwestern Iowa, they gamed out possible nominees in the Democratic presidential primary. Mr. Johnson, a 27-year-old law student, said the large field was a great equalizer, and “if at the end of it we get an old white guy, someone who represents the status quo, it’ll be because they’ve proven themselves.” Ms. Neal, a dental hygienist, made an agonized face at Mr. Johnson, her boyfriend. Wouldn’t something be lost, she asked, if the historically diverse slate...