Standing nearby, on a ceremonial balcony overlooking the West Wing, was another reporter, David Brody, waiting for a private interview with Vice President Mike Pence.

Mr. Brody, the chief political correspondent for the Christian Broadcasting Network, was not there to inquire about porn stars. It was the National Day of Prayer, and Mr. Brody asked the vice president whether he was tired of defending his anti-abortion views amid “potshots” from comedians, and whether prayer was “alive and well in the White House.” He inquired whether Mr. Pence would attend the opening of the American embassy in ...