I'll go along with you on the invasion: good riddance to Saddam, his two wicked sons, and the whole bloody Baathist regime.
The problem with a full-scale invasion in 1991 is that the UN mandate for war said "liberate Kuwait." It didn't mean going on to Baghdad and dealing with Saddam's regime. However, there were contingency plans for just that mission: if Iraq had used chemical weapons against U.S. or other Coalition Forces. Both SecState Baker and SecDef Cheney vowed that the Iraqi leadership would be held personally responsible in that event, and slagging Baghdad with a B-61 nuke (max yield: 500 Kt) doesn't do that: after all, Saddam and his inner circle may be out of the city. Thus, the likely response is that once Kuwait and Southern Iraq are clear of Iraqi forces (cities like Samawah, An Nasiriyah, and Basra), XVIII Airborne Corps, VII Corps, and I MEF reorient to push to Baghdad. IMHO that's the only reason in 1991 to go all the way.
There is an alternative: to quote Gen. Chuck Horner, CINC-Central Command Air Forces, said after the war that "We did not try to kill Saddam Hussein. But we bombed every place where he should've been." Getting Saddam in an air strike or cruise missile attack is more likely. And the slime-licker got away from a strike: he used a converted RV as a mobile command vehicle, and was caught on the Baghdad-Basra highway in late Jan '91, by two F-16s. The F-16s destroyed the front vehicles and the rear ones, trapping the RV and a few others in between. They called for backup air, but they had to leave due to low fuel. When additional air did arrive on-scene, the RV was gone. Chances are, that was him. If any of Saddam's post-capture interrogations dealt with this incident, it's still classified.