I think one needs to be careful in distinguishing between the ideology of militant Islamist leaderships, and the sometimes rather different issues and ideas that motivate a great many of their followers and supporters.

In Lebanon and Palestine, very, very few of those who voted for Hizbullah or Hamas bought into the notion of establishing a global Caliphate (indeed, in my experience even many Hamas cadres don't). Rather, they are motivated by nationalist (or, in Lebanon, sectarian and even class) factors, as well as immediate political grievances—to the point that both parties play down their Islamist ideologies at election time.

In Iraq, we know that most of the people who have been shooting at coalition forces are angry Muslims, but not militant Islamists (although the conflict may push them in that direction).

In Afghanistan, the largest share of the Taliban support base very localist, even parochial Pashtun tribes who don't like lots of (non-Muslim) foreigners, and are deeply suspicious of the Karzai government.

These are all sets of grievances that have been mobilized in the past by a broad array of ideological messages, from Arab nationalism (in Palestine, Lebanon, Iraq), communism (among some in Iraq and Lebanon), the nationalist new left (ditto), or tribalism (notably in Afghanistan).

Given this, even if one could "defeat" leadership ideology, it would leave many of the subgrievances intact.

What contemporary jihadist groups have done, however, is to project a message of devotion and success, with AQ having set the "gold standard" on 9/11, and Iraq having done almost equal damage in terms of sending a signal of (perceived) American weakness and Islamist success to proto-supporters.

There is a parallel with the Cold War here too--the successes of the USSR in the developing world had a lot to do with a combination of anti-colonialism, legacies of past Western involvement, social inequality and underdevelopment, the benefits offered, and the apparent success of the USSR (in the 1960s and 70s at least) as challenger. While there were many third world socialists, I'm not sure many outside Indochina really bought into the core ideas of the Moscow (or Beijing) communist model.

Ironically, I think in the long term it would have served our cause better if we had landed in Afghanistan and crushed the existing AQ infrastructure, but left the Taliban in control. Then we would have actually had an instance of an "Islamic" state to show to any Muslims attracted by the idea.
..although Saudi Arabia, Iran, and (at times) Sudan, could also make claims to being an "islamic state" and that hasn't made much difference. Moreover, Afghanistan is generally considered such a peripheral backwater of the Muslim world that (Pakistani Pashtuns aside), very few Muslims pay it much attention unless it has Soviet or US troops in it..