The inside story of how the producers of “The Apprentice” crafted a TV version of Mr. Trump — measured, thoughtful and endlessly wealthy — that ultimately fueled his path to the White House.

Late in the summer of 2003, a team of television producers stepped off the elevator on the 26th floor of Trump Tower eager to survey the set of their next reality show. After years filming “Survivor” in jungles around the world, training cameras on exotic spiders and deadly snakes to evoke danger, they came looking for a different set of sensory clues, the tiny details that would convey wealth and power.

Right away, they knew they had a problem.

The first thing they noticed was the stench, a musty carpet odor that followed them like an invisible cloud. Then they spotted scores of chips in the finish of the wooden desks and credenzas. The décor felt long out of date, making the space seem like a time capsule from when Donald J. Trump opened the building early in his first rise to fame.