“I have some fans in here,” Mr. Scaramucci said with a toothy grin as he took the stage. “Who knew it was going to happen, right?”

For a time it was an open question in Washington whether people in the Trump White House would be able to compete in the free-for-all cash grab of book deals, speaking engagements and TV network contracts that come with proximity to presidential power. Eight months in, the tradition appears alive and well for the Trumpites, who have proliferated on daytime talk shows, late-night shows and attracted millions of followers on Twitter.

That was evident at the Emmy Awards on Sunday, where Sean Spicer, who served in the West Wing as a perpetually beleaguered press secretary, wheeled out a little podium onstage to lampoon his past assessment of the crowd size at Mr. Trump’s inauguration.

Even Stephen K. Bannon, President Trump’s former chief strategist who has returned to the helm of Breitbart News, took time to drop by “60 Minutes” to indulge his own myth as a bomb thrower — or, in his words, a “street fighter.”