Here is YangMills's answer, so I can accept it (after a several-day waiting period enforced by the software):
The class $\cal{R}^*$ of surfaces considered by Zelditch excludes all Zoll surfaces of revolution, because of the "simple length spectrum" hypothesis (page 2) which is never satisfied for Zoll surfaces of revolution. For example it implies that the length $2L$ of "meridian" geodesics cannot be equal to the length of all other geodesics, while in the Zoll case all geodesics have the same length.