After listening to the first few minutes of the San Diego Symphony’s “Star Spangled Pops” concert with his family July 3, Republican National Committee head Reince Priebus bolted to the airport for a red-eye flight to New York.

Donald Trump and his wife, Melania, were hosting Indiana Gov. Mike Penceand Iowa Sen. Joni Ernst in separate tryouts for the vice-presidential pick, and Mr. Trump wanted Mr. Priebus’s input in a hurry.

The...