Where does it state that COIN requires a functioning government? If you buy into Bob's World's arguments then the conditions were set for insurgency because the State wasn't functioning well to begin with.
I think Karzai is more than a Mayor and he extends control beyond Kabul, but as you state he doesn't control a lot beyond Kabul. If it isn't an insurgency what is it? A civil war? If that was the case then the doctrinal approach (I'm not saying this is right) would be peace enforcement.
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