True, but the Kurds were (still are, I think) pro-American and while the Shiites don't have much use for us, they can count. In a democracy, Shiites will make-up the majority of the electorate. The Shiite plan was to simply wait us out and take over. And if we killed a few hotheads like al Sadr while we were at it, so much the better.
Don't be fooled by the alliance with the Kurds. There are different tribes with different loyalties. But in the long run, Kurds are only loyal to Kurds.