Senior White House advisers for more than a year have aggressively stage-managed President Biden’s schedule, movements and personal interactions, as they sought to minimize signs of how age has taken a toll on the oldest president in U.S. history. The White House has limited Biden’s daily itinerary and shielded him from impromptu exchanges. Advisers have restricted news conferences and media appearances, twice declining Super Bowl halftime interviews—an easy way to reach millions of voters—and sought to make sure meetings with donors stuck to scripted pleasantries. Senior aides dismissed travel suggestions over worries the president didn’t have the stamina for them,...