WASHINGTON—Army chief of staff Gen. Ray Odierno issued the marching orders in the fall of 2013. Some of the Army’s brightest officers would draft an unvarnished history of its performance in the Iraq War.

A towering officer who served 55 months in Iraq, Gen. Odierno told the team the Army hadn’t produced a proper study of its role in the Vietnam War and had to spend the first years in Iraq relearning lessons. This time, he said, the team would research before memories faded and publish a history while the lessons were most…