I actually think Bush senior should have chased Saddam all the way to Baghdad when he had him on the run. Probably would have saved a lot of Iraqi citizens lives who did die at the hands of the dictator. When Bush junior tried, he had the likes of Rumsfeld and Cheney advising him, and we saw what a disaster that became.
Cheney was Bush Sr.'s Secretary of Defense. Paul Wolfowitz was Undersecretary of Defense under Cheney for Bush Sr. Rumsfeld, Cheney, & Bush Sr. were old buddies who served together under Ford & Nixon.
Bush Sr was a holdover from the Reagan years, as his VP. But everything changed with the end of the Cold War in the early 90's, a "new world order" as Bush Sr. famously called it. Then you had those mentioned above who served under Bush Sr. formulating a brand new kind of (neo) conservative foreign policy behind the scenes during the 90's Clinton years. A new more aggressive US foreign policy, spreading liberalism through regime change & nation-building, with a now-unmatched global military empire free from Soviet interference. During the Gulf War, US conservatives weren't as keen on nation-building, & had a bad experience with insurgents in Afghanistan.
Bush Sr. seemed like a bit of a limp noodle, a bit like his son Jeb. Barbara wore the pants in that family. He was more of a placeholder between Reagan & his son, who both had much more impact.