They came by the dozens in luxury sedans, black Ubers and sleek helicopters. As they did each August, General Electric’s most important executives descended on a hilltop above the Hudson River for their annual leadership gathering.

Just an hour’s drive from New York City or a short flight from Boston, Crotonville, N.Y., is the home of GE’s management academy, famed for culling and cultivating a cadre of leaders the company saw as its most valuable product.

Crotonville...