Just before 3 p.m. on Dec. 18, a surprising visitor showed up at Boeing Co.’s military aircraft facility in San Antonio, Texas: First Buddy Elon Musk.

Elon Musk has taken on a new role as the First Buddy: monitoring the progress on the long-awaited revamped Air Force One jets. The new Air Force One planes, made by Boeing, are currently $2.7 billion ...