I was under the impression that the Gulf War was already seen as a great victory and the height of Bush's popularity. Am I wrong about that? I don't think deposing Saddam would've made Bush any more popular in the long run, especially since the ensuing chaos of a leaderless Iraq would be a concern that would probably overshadow whatever shallow patriotism people were enjoying during the Gulf War that happened in our timeline.
Keep in mind in 1991 we would have had 750k troops on the ground and not 150k which makes it easier to stamp out an insurgency. Also keep in mind that the people who made up much of the original military were former members of Suddam's military as in 2003 we went straight for Baghdad instead of trying to destroy their military first while in 1991, we basically were on the verge of completely encircling the entire Iraqi military so if we continued it is very likely we would have literally destroyed their entire military.
I believe this was the reason we did not do this as killing more than half a million Iraqi soldiers while losing 200 of your own would be considered a massacre especially given that we would have done that after already completing the state mission and would have created a horrible precedent for the future too. President George HW Bush unlike President George W Bush was able to see the bigger picture and ultimately made the right decision by not taking out Suddam in 1991.