"Simply that we fail to understand the culture and the "drivers" of Arab and islamic peoples. (and you can add Asians, Europeans, Hispanics, and Africans to that!)"

I think you are wrong. There several reasons are wrong. First and foremost is how are you going to do this- at practical level. Who is going to teach this stuff. Fuck Iraq, why don't you teach people about America culture. Or in your case Aussie culture. You can't do it. Maybe you could teach Arabic that would a good idea- but still this would be massive job but comparitively easy. Now, using Iraqi ex-pats in America would good idea- they will know more than years and years of trying to teach this stuff in some classroom. And as far as I know we did this.

But I'll define the mess. The mess is identifing the leadership in Iraq- the Iraqi living there who are actually running the joint. Elections are nice, and certainly get around doing this. But FIRST find out who is running the place. And I don't mean Saddam's govt, mostly I mean who is running the place despite the bad leadership of Saddam and this socialistic govt. Now we did this in the Kurd area, because the Kurds were our allies. So instead of making new leadership, the focus should been on identifying the leadership that was working- strengthen that. Then find leadership that isn't working. The leadership that isn't working- those trying kill people in order get people to listen to them, you try to salvage leadership or that isn't working take them out- kill them or imprison them.
So as far as I can tell disbanding the Iraq army and police was a bad idea, but if you were going to do this, you should been better prepared at identify good cops from bad cops thereby allowing to to disband and rebuilt with within a short period of time- a month or two. The problem with disbanding is you disrupting a lot of leaders of Iraq. In other words these were people in which "normal" iraqis would go on to solve a problem they were having.
One more thing, by leaders I mean leaders of small groups and large groups. Iraq is tribal in nature- more so than "modern society" so of 25 million people you might be talking around a million or more people are are these leaders.