RICHMOND, Va.—On Oct. 7, 2008, in the cramped TV room of his modest home here, Marty Bannon watched with alarm as plunging stock markets dragged down his shares of AT&T, the nest egg he built during a 50-year career at the company.

His five children, including current White House counselor and chief strategist Steve Bannon, had often joked growing up that their devout father, a product of the Great Depression, would sooner leave the Catholic Church than sell those shares. The stock symbolized his deep trust in the company...