I had the same thought that the women should go first.
It has some advantages, some already stated, another being that it allows those unfamiliar with the course to become more familiar. One of the great things about Augusta or Sawgrass is that the viewers know the course very well and can compare year to year performance.
But then I realized that there might be some viewer fatigue, and part of the excitement of any golf tournament is the unknown of how the players will adapt to the course. Some of that excitement might be lost, if we've already watched the course for a week.
Therefore, I'm starting to think they just need to have the men's and women's tournament at the same time on the same course!