I think rd95 already said just about everything that I would have said.
Anecdotally, I'm a huge Christopher Hitchens fan, and by necessity of that fact I've thought about this subject a lot. What is it that makes Islam so conducive to horrendous violence? Why is it that in this one particular group we see this particular brand of violence?
Those questions assume a certain type of answer and you've already given away some explanatory agility.
I think what we see behind every society that spawns jihadists is a history of American/Soviet proxy wars, or some other external meddling. If you broaden the scope of your considerations you'll see the same sort of moral/civil decay anywhere that this scenario has played out. The Korean peninsula? Vietnam? To a lesser extent Cuba. Israel/Palestine has all the right ingredients for precisely what you would expect.
So, to join it all up, when you put the screws to small countries extremism is the only chip they have left on the table. The solution to extremism, in my eyes (despite my love of Christopher Hitchens) doesn't rest in Islam. It rests in what I think is the logical place, the place we should have been looking all along; helping people of whatever religion or background develop and maintain functional social/governmental institutions that provide the necessities of life. Get that going in any society and I think you'll see extremism dry right up. In a generation? Perhaps not. But a century of blood isn't always easy to wash away. The moves we've made with respect to Iran and Cuba look very refreshing if you ascribe to the conclusions I've set out in my thesis; if there is a cure for extremism it will be in the form of a secular justice that eventually redeemed France, Germany, Japan, or any other country you'd like.
I think the best answer to your question is that the problem with Islam is that it got stuck in the middle of an international conflict between foreign super powers.