Pete Buttigieg lost his primary bid on Sunday night when he dropped out of the Democratic presidential race, yet there’s little question that the former mayor of South Bend, Ind., won his personal campaign to become a national political figure.

Mr. Buttigieg had tried before to become a power broker in national politics. His early efforts failed: In his race to become the chairman of the Democratic National Committee, Mr. Buttigieg stepped aside before the vote was tallied to avoid a potentially low vote total.