Why is it worth it to build up a new government in Iraq if it means it will take longer than a decade? Why do we care? Sell me on this Harm, because clearly you're drinking the kool-aid.
Because without a strong government, these extremists will continue to exist. We can ignore them all we want, but eventually they will through a big rock at us. Unfortunately there is no magic bullet, and there a tons of interconnected problems. These groups are like weeds, hard to deal with, and even harder to deal with once they grow over time. At the same time they want us to come fight them, they are trying to bait us beause then they can try and justify their actions by calling us imperialists trying to take them over.
History: Jews, Christians, and Muslims have never gotten along. Because of this, we have had many holy wars, crusades, and even attempted genocides. You have peaceful majority, and a violent minority in each religion. These extremists are fueled by history of hatred, and with hatred you can win people over. Sometimes we can unintentionally create these extremists. For example civilian casualties accidently caused by us can turn the people we are trying to help against us. Look at the Christian preacher who wanted to burn all those Qurans, his hatred came from the actions of the Islamist extremists. Bin laden attacked us because we were aiding Israel and occupying friendly arab near Iraq incase we ever needed to invade them again. The Israelis have dropped cluster bombs and shelled muslim civilian populations in response to attacks on them, which leads to hatred on the jews from both christians and muslims.
Basically if you can some how make everyone forgive each other and let go of thousands of years of history, this would be the biggest mile stone in bringing long term peace to the middle east. Some form of tolerance where everyone can cohabitate without conflict based on what you are.
Another issue though is trust. You take a man in the middle east, give him power, then he becomes the newest dictator. Look at Assad in Syria, or Saddam in Iraq, or the previous president in Iran. All of them corrupt starfishs. Look at Iraq which now has many corrupt and incompetent politicians. How can you trust them? I'm of seeing stuff like in this whole video happen:
https://www.youtube.com/watch?v=Ja5Q75hf6QICreating a Kurdish state is promising, However we risk Turkey leaving NATO. There is also no guarantee of Kurds and Iraqis getting along, nor Christian and Muslim populations. Plus this could lead to more crazies rising up because some Christians have taking over part of their holy land.