He looked at the globe and saw that one and a half billion people were Muslims and even a relatively infinitesimal percentage of them means that millions can be committed to wage murderous, even suicidal, war against the United States. That in a context in which it required only 19 militant Muslims to bring down the World Trade Center and bomb the Pentagon.
Ultimately, I think Bush understood that such a war had to be won by Muslims themselves reforming themselves. At least that was at the time a strategy one as president of the United States dared not throw away with loose rhetoric.
So Bush chose to wage war against terrorism rather than against militant Islam because he feared that 1.5 billion enemies were worse than a minority of Muslims committed to terrorism. As I said in my original post, this strategy has obviously failed but it was not an unintelligent effort.
The support of moderate Islam is precisely what we need in this war against militant Islam. Without it we are going to suffer setback after setback, incur casualty after casualty, and spend ourselves bankrupt. That does not necessarily mean that we have to kowtow to the inflated sensibilities of Muslims who are, as I pointed out in my original post, often thoroughly irrational.
