Culture from barb camps should stay. It makes clearing them that much more rewarding, both tangibly and mentally. If you're looking for an experience-from-barbs theme, maybe you could give a stack of experience to a unit that clears a camp?
Farming camps is never a good idea. It's gamey, dangerous, begging for some other civ's pathfinder to swoop in and clear the camp, and worst of all, it's boring as hell. Never do this and never implement mechanics encouraging this. Correct me if I'm wrong here, but I'm pretty sure that the rate at which an individual camp spawns barbs is similar to a new camp spawning after one is cleared. Since camps spawn with two barbarians you'll get more barbarians to fight this way anyway, even ignoring the rewards from clearing the camp.
It'd probably be too much or too late, but what do people think of Authority's opener giving some kind of scouting bonus? Reveal nearby cities in some way, increase military or recon sight, something like that.
Authority is going to be swingy and map-dependant to a degree no matter what. The tree is based on being aggressive against your neighbors, and is therefore dependant on who those neighbors are, moreso than any other policy tree. I don't think you can fix that without changing the tree's identity outright. So maybe there should be something to help the player determine if this really is an effective Authority start and to keep going, or to bail and go Progress or Tradition?
It's a little beyond the scope of the Authority tree, but my god, I love absolutely everything in this post. Genuine mercenary work, consequences for under-performing, building a stable alliance with a peaceful civ as their go-to attack dog, inevitably turning on them and tearing them apart when they stop paying up, get too cocky, or when you get so strong you just don't need them anymore. Yes, yes, give this to me, inject it into my veins.