ATHENS, Greece—After three excruciating overtimes and the most epic game of his life, the 18-year-old kid bound for NBA superstardom was broken. He couldn’t bear to look at the scouts in the courtside seats. Or the riot police on the baseline. Or the mob of fans lighting flares inside a dingy gym.

Filathlitikos, the only pro basketball team that Giannis Antetokounmpo had ever known, was about to lose its unlikely shot at promotion from the second division of Greek basketball in 2013. As red smoke filled the gym, Antetokounmpo...