Kind of. If you say "anything but" you are stressing that fact that it can be any possible thing except for the one mentioned. 'Everything but' does not quite have the same subtle meaning. You are saying that an event happens more often than it does not happen. For example, a very good sports team could be said to 'win more often than not'. Meaning that they win more often than they lose.
I think the best solution involves creating a trust from all of the oil money that directly goes back to the citizens of Iraq. Much like Alaska did for awhile (citizens of Alaska did not pay state income tax and actually received money back from the state). The quicker money gets flowing into their economy and people realize how great their lives can be, the quicker the regular citizens turn on the few who are causing all the problems.