MANILA—At each stop on his 12-day, five-country Asia tour, President Donald Trump emphasized one aspect of his visit above all else: his personal rapport with the heads of state who hosted him.

He golfed with Shinzo Abe, complimenting the Japanese prime minister’s accidental backward tumble into a bunker as graceful. He convinced a reluctant South Korea President Moon Jae-in to meet him at the North Korea border, a visit that thick fog ultimately forced them to cancel. And he engaged in a favorite pastime—making jibes at the...