Before he stepped into the role of chief executive at Gap Inc. in February 2015, company veteran Art Peck believed he had hit upon one major problem at the slumping apparel maker: Its creative directors had too much say over everything.

By the time Mr. Peck took the top job, the creative director of the retailer’s Gap brand had already been dismissed. A few months later, the company’s Banana Republic chain eased out its creative head and eliminated the position.

The...