There really should be three Iraqs; Shia, Sunni and Kurd. The three will never be able to co-exist without a Stalinist dictator like Hussein keeping them in line.
Reality would be nice if it weren't so complicated.
Sure, let's just sit here like the "pirates who don't do anything" and say that three little Iraqs would be so much better.
Except that three little Iraqs might quickly turn into The Land Of Unintended Consequences.
1) Iran covertly or overtly annexes Shia-stan, occupies Basra and cuts off gulf access.
2) Sunni-stan devolves into a new hardcore Muslim Brotherhood terror training headquarters, unfettered by American military intervention.
3) Kurdistan becomes a success story just long enough to provide an example to Turkish Kurds that the best idea for them is to revolt against Turkish government and secede, bringing American troops into a hot war between the Kurds and the Turks.
Just because Iraq was an "artificial construct" of British imperialism doesn't automatically mean it's the worst option.