A cynical possibility is that Cheney wasn’t positioned to make a profit off of an Iraq invasion in 1991. So he spent a decade reformating his corporate interests and then kicked off the war when he was better prepared for it.
George Bush the Elder believed first and foremost in stability. In his mind, ONE nation headed by ONE government (or even by ONE tyrant) was always preferable to chaos.
Remember, Bush wanted Yugoslavia and the USSR to stay together. He thought one USSR ruled by Gorbachev was much better than 20 republics, each with its own nukes.
Bush probably figured that one Iraq under Saddam was preferable to an Iraq in chaos or to three mini-nations (with, say, one ruled by an Iranian puppet, one ruled by one of Saddam’s old lieutenants, and one held by several different squabbling factions of Kurds).
I don’t think that Iran would have had a big hissy fit about the US going in and occupying Iraq. See Iran-Iraq war. No love lost.
So what, there was no love lost in 2003. Iran would of made any US attempt at controlling Iraq a quagmire, and why not? It’s the principle enemy of the Iranian government, which has removed a dictator they perceived as the main blocking point of Shia ascendancy in Iraq.
The majority Shiites take over and that gives Iran influence over what happens with Iraq’s oil. The Kurds take or try to take or take control of the oil in the North and get Kirkuk back. This drives NATO ally Turkey nuts. Sunnis would have been driven out of the Baghdad area as they were to a certain extent from 2004 through 2007. Al Qaeda is Sunni. Iraq would have become a dysfunctional state worse than under Baathist control where Al Qaeda would have gained a foothold just as they did in 2003.