Brendan Carr loves a good luggage metaphor.

When he was a law student at Catholic University in the early 2000s, Carr heard a talk from Michael Powell, then the chair of the FCC under George W. Bush. “When opportunity knocks, most people don’t have their bags packed,” Powell told the audience. Carr was so energized by the line that he made it his own rise-up mantra and now recites it when speaking to law students himself.

The 46-year-old has been unloading a lot of baggage lately. Ever since Donald Trump bumped Carr up to chair of the FCC in November (after naming him a commissioner during his first term), Carr has attacked the country’s media, entertainment and even tech giants with a cool fury, threatening their business and, critics say, attempting to bully them into more favorable coverage of the president.

“Americans no longer trust the national news media to report fully,...