In the spring of 2014, Las Vegas tycoon Sheldon Adelson floated an offer to David Solomon: come run his casino empire.

Mr. Solomon, a senior Goldman Sachs Group Inc. executive, had been Mr. Adelson’s banker since the 1990s. He had honed his craps game on weekend jaunts to Atlantic City, N.J. He was 52 years old and seen as a long shot to become chief executive of Goldman.

But...