It’s hard to think of Steve Bannon as a low-profile guy. He has garnered about as many headlines over the past week as Donald Trump—no small feat. He is the executive chairman of the hard-right Breitbart News, among the most aggressive voices online, its website an attack machine against Democrats and “establishment” conservatives. President-elect Trump chose Mr. Bannon this week as his chief strategist and senior counselor, a slot usually filed by someone eager to play a presidential surrogate on TV.

Yet Mr. Bannon—who joined...