We can all bemoan "could a", "should a" with Iraqi and its aftermath, but it won't change anything, but more importantly we don't know if it would have made things worse or better.
Using that scenario is a simple argument and its a lot more complicated. Control of a population can be extremely complicated and having a foreign force controlling a local force that was once loyal and possibly same sect as the previous government was a recipe for disaster.
Everything depended on how the Shia was going to react. We had enough problems dealing with the Sunnis and the relatively small amount of Shia troublemakers loyal to Sadr. The last thing we need was Sadr getting a lot more support.
The effect of that could have been massive unrest amongst the Shia population, and the only way to control that would have been to resort to brutal tactics via our proxys.
The end result is we can take our favorite political scapegoat standpoint, but its as relevant as blaming Iraqi not going to plan because Turkey wouldn't let the 4th ID in according to the original game plan.