Pete Buttigieg seeks to become the first Democrat since 1924 to win the presidential nomination without serving as governor or U.S. senator. He’s pitching his experience as a small-city mayor as just what the country needs: “We have to start getting Washington to look like our best-run cities and towns, instead of the other way around,” he said last weekend in New Hampshire. The argument was enough to carry Mr. Buttigieg, 38, to a close second-place finish in Tuesday’s primary.

Mr. Buttigieg can point to some successes in South Bend, Ind. But his eight years of leadership, which ended last month, left the city of 101,860 neither one of America’s best-run cities nor a model of political comity. He came into South Bend like he has the presidential campaign, striking gauzy, aspirational themes. He rode the national recovery to modest economic gains but ran into trouble on the hard urban problems of crime and racial politics.

He took over a city that had long been in decline. South Bend’s population peaked at 132,445 in 1960, more than two decades before his birth. In 2011, the year he was elected, Newsweek listed South Bend as one of America’s top 10 “dying cities”—a onetime manufacturing hub that had never recovered from the shuttering of the Studebaker auto plant in 1963. “What is particularly troubling for this small city is that the number of young people declined by 2.5% during the previous decade, casting further doubt on whether this city will ever be able to recover,” Newsweek wrote.

City Hall was about to undergo a generational change. Mayor Steve Luecke was stepping down at 61 after 15 years in office. Mr. Buttigieg, then 29, jumped into the ring. “I think the fact that I’m young is precisely why I can move the ball forward and break some old habits,” he said. “The fact that I’m not currently an elected official, I think, helps me.”

“As the youngest of the candidates, 29-year-old Buttigieg exudes energy and enthusiasm,” the South Bend Tribune wrote in an editorial endorsing his candidacy. Although he was the least-known of the Democratic primary candidates, he raised the most money—enough that one opponent accused him of “trying to buy an election.”