MEXICO CITY—Every weekday morning at 7 o’clock sharp, President Andrés Manuel López Obrador stars in his own one-man show.

As dawn breaks over this capital city, the 65-year-old takes to a pulpit in front of the press, where he answers questions and holds forth in a folksy style about everything from the price of gas to whether drug lord Joaquín “El Chapo” Guzmán got too harsh a prison sentence.

A...