The bourgeoisie's media and cultural tools are used to create a false consciousness among the working class. That's why many of us vote against our own interests.
If an Iraqi liberal democracy had been created, I have no idea what it would look like. It's very hard to imagine, as that region has a very weak history of democracy. However, we could probably predict that it would end up looking something like Turkey's system, with Islam being much more present in government. It's likely that the power struggles between Sunni and Shia Muslims would be represented in a parliament or house of representatives (whatever legislative assembly they create), and would, as in our timeline, be greatly inflamed by the American invasion.
I'm happy to find sources to counter it, but I remind you that the burden of proof is on you. You're the one who claimed that billionaires will flee NYC.