WASHINGTON—Soon after taking the helm as President Trump’s third national security adviser, John Bolton installed a gold-framed painting of President George H.W. Bush sitting at the Resolute Desk in the Oval Office, surrounded by his vice president, national security adviser and the secretaries of State and Defense as they discussed plans to invade Iraq in 1990.

The painting, which hangs above Mr. Bolton’s West Wing desk, serves as a symbol of his approach to the job, seven months in. He favors high-level, small group, centralized...