Worth noting that the Qur'an isn't the only text in Islam, as much (or even most) of the religion's practices are actually sourced from various hadiths. Hadiths are written works attached to different sects of Islam, written after Muhammad's death, that were mostly oral records of his life, lifestyle, sayings, etc. One of the results of this is that, since the guy lived in the 600s, a lot of the stuff in there is typical of life for Bedouins in the 600s. It's the basis of a lot of the ritual and social traditions, as well as Shariah law. Many people also don't realize that Islam has such a huge body of theological works that were developed later, beyond the hadiths or Qur'an, much like Catholicism.
You could make an argument that fringe groups have enough material there to find justifications for the kind of revolutionary religious extremism that has been common in the last 40 years, but that's not the cause of that violence, and the religion itself really isn't the problem. The cause is that after WWI Britain and France divided up the Ottoman Empire along arbitrary lines with no regard whatsoever for ethnic, tribal, or religious differences of the people living there, and then went about exploiting the region like every other colonial mandate or puppet state. With Russia and then the United States joining in later, and then the dismantling of the European colonial empires, the region was left lacking governments legitimate or capable enough to remain sustainable without collapsing into military dictatorships (as was par the course for almost every post-colonial state). If that weren't bad enough, like everywhere else on the planet at the time the CIA and the USSR were rigging elections, supporting rebels like the Taliban, performing coups and so on, again, with no regard for the people there.
But if you want to look for the largest reason why this became a religious struggle (or rather a religiously motivated revolution), and not a political one, it's gotta be the creation of Israel. Britain's colony Palestine, another one of those arbitrarily drawn up provinces, lacked local political leadership the British were willing to pass it off to. Of the European empires, Britain was probably the best about trying to ensure a stable successor government, but the results (as evident from Africa and the Middle East) rarely saw long term success (or for any longer than 3-10 years, really). Palestine was exceptionally bad, given the extreme religious tensions, and after all the revolts, massacres, civil wars and so on, Britain just said "fuck it" and let the UN go through with a half-baked plan to create a new Jewish state there, completely ignoring the majority Arab disapproval. For any oppressed and discontented Muslim out there, all they need to do is look at what happened with Israel to have a glaring reminder of everything that's gone wrong (and continues to go wrong), and a very real reason to distrust the West. Obviously, most moderate Muslims in the region aren't about to get violent over the issue, but enough disenfranchised radicals are that it's a major contributing factor to the violence there, and it's not difficult for moderates to become more sympathetic to extremist groups, when, say, a drone strike kills half your extended family.
There are of course other factors, like the US's oil interests, the dubious alliance with the Sauds, gradual weakening of the dictatorships with the end of the cold war, etc. But with the wars between US-backed Israel and its neighbors, the continued lack of a resolution between Israel and Palestine, the Soviet invasion of Afghanistan, 9/11, the US invasion of Iraq, the US invasion of Afghanistan, US bombing of Yemen and anywhere else they please, and so on, it's an environment that has been destabilized for so long that it would be a miracle if there wasn't constant fighting, religious or otherwise. Going at it from a religious angle just doesn't make any sense, as you can find remarkably similar situations that emerged in Africa, South America, and Southeast Asia after the colonial period, some of which continue today.