The only question of real value is whether the threat posed to U.S. interests by Baghdad could be contained or eliminated without amplifying the greater threat from Tehran. So far the answer appears to be "no".
The problem with Iraq wasn't the invasion or the reasons for it, but our game-plan once we had control of the field. We should have turned Iraq into a vassal state, setting up a U.S. military-run government under which we provided security for the Iraqi people, establish Western institutions, introduce and protect religious diversity and train both civilian and military personnel in Western administrative techniques and values. The Iraqi people, precisely because they are Muslim, are unsuited for democratic institutions and the instability of the current regime and its susceptibility to becoming another Iranian puppet state are proof enough that we don't understand the fundamental nature of Muslim theology, Arab culture or the unsuitability of either for self-rule as a peaceful member within the community of nations.
We had an opportunity to build in Iraq a prosperous, peaceful, friendly ally and we blew it. It's only a matter of time before it becomes more obvious. It isn't that the goal was wrong so much that the administration severely underestimated what it would take to get Iraq there. That elections are held, as is demonstrated time and again, is neither proof nor guarantee of the wisdom of the electorate, nor does it usually result in good government.
Iraq didn't need democracy. It needed a de-Islamification.
You make a lot of assumptions and generalizations about Islam. What about the rest of the Islamic world that sees successful democracies and Turkey or Indonesia?
Also you over estimate our ability to turn around into your "ideal" country, the matter of "civilizing" a people isn't that simple nor is it that quick. History is full of examples of one group of people trying to tear apart another group from its history, culture, national identity, in fact its very essence. Set up a puppet government? Sure why not, thats fairly easily, but getting the people to actually follow along with that government is an entirely different matter. There is no way in hell you are going to take even the youngest Soldier, when we are talking about training the military, and remove his life previous to his introduction into the Army and teach him a culture very different from his own.
Islam is no more opposed to democracy than any other religion, and its not a culture's religion that gives it the ability to be democratic or not its their people, their history, and their culture. Simply because Islam is part of their history, culture, and identity, which is a major hurdle for democracy, does NOT mean Islam is the only problem. For example in the Bible there is no mention of any of the democratic and liberal theories which were created and developed throughout centuries to finally manifest in the types of democracies we see today. Religion is a big part of their lives, like its a big part of ours, but its not the only defining feature of Muslim's existence, just like we don't stone people for adultery even though its in the Bible, so can Muslims ignore some of the more gruesome portions of their holy book if their culture allowed it.
If Western culture didn't have the same kind of great thinkers that it did, we may be just like the dictatorial countries of the Middle East.
By the way if you want an idea of how hard it would be to de-Islamify Iraq, think of how hard it would be to de-Christianize you. The only way to de-Islamify Iraq is to depopulate its country.