WASHINGTON—The confirmation of Secretary of State Mike Pompeo on Thursday installed the final piece of President Donald Trump’s revamped foreign policy team, transforming his inner circle from a disparate group of strangers to a band of Washington insiders who have accrued more than 115 years of government service.

As he lobbied the Senate for his confirmation, Mr. Pompeo, a former House lawmaker who most recently ran the Central Intelligence Agency, marketed himself as the official who could unify Mr. Trump’s foreign policy...