What I think happened was that Bush saw the long term problem in the Middle East, not just in Afghanistan, but throughout the region. He looked at the whole picture and decided that Saddam Hussein was in the strategic center of the region, not just the geographical, but ideological center. He wasn’t part of the Al Qaeda movement because the goal of Al Qaeda is an Islamic caliphate which would displace Saddam as the supreme ruler. Yet, Saddam was equally ruthless and cruel to his own people and a major financial supporter of Islamic terrorism because he hated westerners, particularly Americans and Jews.
Bush wanted to establish a long term American base in Iraq, from which the US could monitor and influence Middle Eastern threats to US interests at home. I think the idea was to try to keep putting out Islamic brush fires before they jumped the Atlantic.
I think you got part of it
President Bush revealed a vast misunderstanding of the Middle East problem when he referred to Islam as “a religion of peace”.