I was paying quite close attention during 2002-2003 and the neoconservative belief was that Iraq would become a shining, neutral if not pro-Israel, light unto the other Gulf nations. Iraq was seen as an easy military victory in part due to the ease of removing them from Kuwait earlier. Also, in many ways Iraq was the most secular country in the region even if not presented that way in the media for obvious reasons. And it was next door to Syria and Iran. Very convenient.
What’s not to like? From there to eliminating Iran’s militarily significant threats would be easy.
At first everything went swimmingly in spite of not finding WMDs though it became obvious early on that wasn’t high on anyone’s agenda outside of the media that kept bugging Rumsfeld. It wasn’t, after all, the real reason to invade Iraq. There was even the piece in WaPo titled “We Are All Neoconservatives Now.” There were pressures to “do” Syria and Iran and some of those whispers made it out into the public. More likely than not to pressure Bush to not stop now.
Then reality set it. Turned out Iraqis weren’t thrilled to be occupied and Bush started to doubt the wisdom of further expansion. It was reported at the time Rice and Bush decided to delay further expansion until things stabilized. Stabilize they didn’t. Just before Bush’s term expired Cheney and the neocons put a full court press on Bush to bomb Iran. My guess is he was more than pissed about the whole thing given the background of his old man and some of the old guard. But, of course, Jeb’s political ambitions, like Hillary’s muted anything from either W or Bill. But they know the deal and know what they must do to remain viable.
For the neocons, it’s always been about Israel. Anyone that’s paid any attention to them understands that.