This was because of oil revenue. Most of the oil is in the Shia south and the Kurdish north and the Sunnis don't have much.
If the country were partitioned, the oil revenue would also be partitioned, leading to endless war over the oil money.
The US insisted on and the Iraq Constitution contains the oil revenue sharing language.
But problems arose quickly. In 2007 Hunt Oil Company announced that they would deal direct with the Kurds. Other companies also announced but they were all small companies and the oil still had to be shipped south to the Persian Gulf allowing the Iraq central govt to collect the revenue and distribute that revenue according to the Constitution.
To deal with this problem, the Iraqi Central govt developed a policy that said any oil company that dealt directly with the Kurds could not participate in any other oil project within Iraq.
But then the problem got worse.
In 2012, Exxon announced that they would deal direct with the Kurds. Conoco and Total soon followed with their announcement. Exxon had bid on and been awarded 50% of the huge Quarna field with a Chinese company holding the other half. So Iraq told Exxon that if they were going to deal direct with the Kurds, they would have to divest their holdings in the Quarna field. Exxon began divesting, although they still hold about 10%.
But then the problem got worse.
The pipeline connecting Kurdish Iraq, thru Turkey, to the Mediterranean was completed. This allows the Kurdish oil onto the world market without the shared revenue being deducted. The Kurds could keep all of the revenue.
If this happens, eventually the Shia will also stop sharing the revenue from the oil produced in the south.
The Sunni, deprived of the shared revenue, will use military force to get their share.
Iraq can have territories of various religious sects and that will NOT affect any oil revenue. Alaska sends each citizen a check for their part in the revenue...that would have worked if our government had pressed for it....when we occupied Iraq. Of course it won’t work now, as we have NO part in the decisions to create a democratic Iraq but considering we opted to get their elections up and running before ironing out the details, it wasn’t a good idea not to make consideration for that, and we would have had individual support if they were getting paid as citizens, then we could have gotten some repayment of our costs to help them by brokering the deal.
Now, a single Iraq would still exist, no one is suggesting otherwise...having had safe territories that were patrolled by their own security would have stopedp some of the sectarian violence...
Territories of Sunni populations and Territories of Shiites, Kurds, etc would also have promoted businesses due to a safer environment to invest time and money with their security keeping out Al Qaeda and other terrorists.
With any opposing factors there can be intermittent violent acts, but it wouldn’t have been the barbaric warmongers running around like mongrels. we see with ISIS..each group would protect their own people.
There could have been jobs, royalty checks on the oil, and a chance to give their children a better life. Not probable now for sure....that ship sailed when we pulled out in 2011.