Invading Iraq was not a mistake. Acting like pussies once we got there was a huge mistake. Had we killed another 250,000 Iraqi civilians and eliminated Tikrit and Fallujah from this Earth, killed Al Sadr and hundreds of others (MacArthur hung 950 Japanese agitators in a relatively tame occupied Japan), stuffed ham into the mouths of every killed insurgent and burned their bodies, forgot about stupid elections until every ounce of resistance was brutally crushed, we would have no problems there right now whatsoever. Where the hell is the rage we all felt on 9/11? Who gives a shit about Arab democracy (as if there ever could be such a thing)? We wanted vengeance and we got Realpolitik instead.
Well, to me, the situation is a little more complicated than your solution would dictate.