From my understanding, women play a huge role in the LDS church, e.g. The Relief Society (which was established in 1842 and has about 6 million members). Even though they don't have the priesthood and as such don't serve as Bishops, Stake Presidents, etc., they still give talks and say prayers in Sacrament Meetings and are in charge of Primary, Young Women's, etc. But I don't understand why LDS women are not able to be Sunday School Presidents. They can be teachers, but not the presidents? What is the LDS doctrine behind that? Why must a Sunday School President be a priesthood holder if they are not overseeing priesthood ordinances? When anyone receives a calling in the LDS Church, they can receive inspiration for their stewards -- regardless of what their gender is, so I don't know why must a Sunday School President be a male, especially if priesthood activities are not involved. Is is mainly based on tradition? Or is their doctrine behind this?
To preface my answer, not all Church policy is shaped by doctrine. Some policies are practical.
That said, I don't know why this is the case, but the handbook does say:
12.2.2. Ward Sunday School Presidency
Members of the ward Sunday School presidency are priesthood holders.
Also, speaking of the ward Sunday School president:
He also oversees the calling and setting apart of other Sunday School workers. He may assign his counselors to call and set them apart.
This, of course, requires the priesthood to do.