Randy Lerner didn’t know much about English soccer in the summer of 2006. But he knew an investment opportunity when he saw one.

Lerner was the billionaire chairman of credit-card giant MBNA, the crown jewel in a business empire passed down to him by his father, Al Lerner, which also included the National Football League’s Cleveland Browns. Financially, the Browns were a small part of the portfolio. But Randy had come to enjoy the trappings of owning a pro sports team—above all, the intoxicating effect of filling a stadium,...