Days after the U.S. presidential election in November 2024, Brendan Carr, a member of the Federal Communications Commission, posted a photo of himself with Elon Musk and President-elect Donald J. Trump at a rocket launch for SpaceX. “A historic day at Starbase,” Mr. Carr wrote in the post, referring to the Texas headquarters of Mr. Musk’s rocket and satellite company. “Another giant leap forward.” Around the same time, Mr. Musk recommended Mr. Carr to Mr. Trump as an ideal leader for the agency.