I think in the right campaign it could be fun. However, I feel getting too statty with organizations can add complexity where it's not needed. Unless you are going to run some sort of interorganizational combat scene, I'd imagine important traits for an org would be the Aspects that define its goals and drawbacks.
If I wanted a real up-in-the-air, political stage, big-picture game with statted organizations, I'd stat the orgs with Aspects and Skills, sure. They'd have some sort of stress track or tracks. Then each session, assuming the PCs were somewhat successful in their goals, it could count as a Maneuver in the overall conflict between organizations. I'm thinking something like Burning Empires or Reign here, where the PCs serve to tip the scales in this overarching political conflict but their efforts aren't guarantors of success in the big picture even if their personal actions server their individual purposes.