I think you’ve correctly identified the crux of the disagreement here. The particulars of ideological beliefs mostly don’t matter. Japan is increasing military strength under a rhetoric of pacifism. The same people who say they would be willing to help the police round up Muslims if the government asked for their help also mostly say they would be willing to help round up “authoritarian followers” (who are defined as people who are dangerous because of their willingness to be aggressive in defense of authority). If you prove one conspiracy wrong, the conspiracy theorist just gives you another because they simultaneously believe many conflicting theories anyway.
People mostly are not self aware enough to understand why they behave the way they behave, and most of the reasons you will hear people give for their behaviour are after-the-fact rationalizations that, if shown to be untrue, would be rapidly replaced by other rationalizations without changing the actions of the person at all. If you want to know why a group of people really does something you look at whether their needs are being met and at the emotions, not at what they say.
Sure, if the justification for killing was Communism instead of Islam you might have some different people involved. You might have a few more or a few fewer people involved. But you take a situation where lots of people don’t have any security - physical, emotional or even food security - and you introduce any kind of rhetoric that says they can get some power back by attacking someone else, and the people will buy it. To the extent that it is believable more people will buy it.
I don’t know if your experience of the world is that most people behave rationally, behave in a manner that reflects their expressed values, and behave in a manner that is consistent rather than hypocritical and contradictory. My experience is that people very rarely do any of those things. I feel like saying that the professed ideology of ISIS is very important to the reasons why they do what they do is putting a level of trust in their word that far exceeds the trust I have in the word of anyone else. I’m not going to assume that terrorists killing people in the name of Islam are rational, reasonable, consistent, or that their expressed reasons are meaningful, since it seems unlikely to me that anyone who is rational or reasonable would be advocating killing, let alone strapping on a suicide vest themselves.
