In a posh suite at the Trump Hotel, Donald Trump Jr. sat next to JD Vance, the new Republican vice presidential nominee, and couldn’t hide his excitement. With a grin, he bragged about how hard he worked to convince his father, Donald Trump, to pick Vance. “I put in a ton of effort—like 1,000%—to get this done,” he said, clearly proud of his role in shaping this partnership.

It was the morning of the vice presidential debate, and as they enjoyed their coffee with a view of New York’s Cen...