1991 Gulf War should have ended with Husseins regime toppled. I remember the reasons given for not going to Baghdad. #1 was loss of support amongst the Arab Allies (which were not needed), and #2 was the taking of Baghdad was supposed to be some "big deal" militarily (which it turned out not to be).
Then we ended up with having to mess around with Saddam crushing the revolt that never was, UN sanctions, no fly zones and food for oil and all that nonsense that did not work, and that dragged on for more than a decade. Fast forward to 2003, and invasion #2 goes in to finish the job. Baghdad falls like a house of cards, and Saddam goes down a hole with a pistol...
I think Bush Sr. was badly advised. Too much "this wont look good on CNN" over what needed to be finished. The blame can be shared by the leaders of the 1991 Coalition as well though, all of them, who didnt have the stones to see it through. If it was going to be "too ugly for TV" (highway of death ect), then they should have never invaded.
The 1991 War had broad international support, there was a clear mandate to do it (western oil, a threat to Saudi Arabia, and the invasion of Kuwait), and the opportunity was squandered. Instead it took 12 years to finish the job properly, dressed up in an unpopular "hunt for WMD", and link to terrorism. Valid reasons, but not as strong as the original mandate was by any stretch.