Zahi Hawass doesn’t like what he’s seeing. Clad in his familiar denim safari suit and wide-brimmed bush hat, the famed archaeologist is standing inside the burial vault of the Step Pyramid of Djoser, a six-tiered, lopsided mound of limestone blocks constructed nearly 5,000 years ago. The huge, gloomy space is filled with scaffolding. A restoration and conservation project, at Saqqara outside Cairo, initiated by Hawass in 2002, has been shoring up the sagging ceiling and walls and staving off collapse. But the February 2011 revolution that toppled Hosni Mubarak—and also ended Hawass’ controversial reign as the supreme chief of all...