The article's main argument is that in order to effectively counter ISIS, the US government must understand them on a theological (i.e. Islamic) basis. At first glance, I thought it was an interesting and sensible argument: let's try to understand the people who work for ISIS from their perspective. But there's an important difference between understanding from "their" perspective and understanding their "Islamic" perspective. True, the article jumps through the necessary hoops of journalistic balance by giving some space to all the other Muslims out there who condemn ISIS as essentially un-Islamic. But this is a mere mention, and the article suggests ultimately that while we should not implicate all Muslims with the actions of ISIS leadership, there is something undeniably Islamic about the organization thanks to its identification with Koranic scripture and theology.
I doubt the authors would commit themselves to such an unsophisticated understanding of religious affiliation, though there are polemicists who are only too happy to make such simplistic claims. The Atlantic argument rests on the idea that the more the United States Government understands the theological underpinnings of ISIS's philosophy, raison d'etre, and organizational structure, the more capable it will be in combatting ISIS. In the article's words, this will make ISIS collapse from its leaders' own "excessive zeal". But it's not clear to me what this collapse actually means. It is surprisingly vague about what collapsing under excessive zeal actually means or what it would look like.
Instead, the writers seem more interested in telling us how this kind of area-specific (indigenous?) understanding doesn't fail, in distinction to competing paradigms. The authors charge the US Government with foolishly treating ISIS as something that it isn't, which leads to vague accusations of tactical "blunders" without giving us examples of what those blunders actually were. I get the impression that the authors want us to think of ISIS not as a terror network (a la the FLN as depicted in The Battle of Algiers, which by the way is apparently used to train counter-insurgency tacticians in the United States), but as an organized theological-nationalist movement bent on hegemony in the Middle East.
But once we start conceptualizing about ISIS as a contender to end the power vacuum in Iraq, it starts to make less sense thinking of it as having some sort of exceptional "very Islamic" character. The alternative to the writers' proposition is that we simply see ISIS as one of many other nationalist movements that have ignited in the past several decades. In that sense, the Islamic aspect of ISIS just becomes yet another instance of a nationalist group looking to create some sort of imagined notion of "pure heritage" on which to found a strong sense of solidarity, devotion to the group, and self-determination. Like the use of archaeology to "prove" that Palestine had always been a Jewish nation, or the Teutonic totemism backing the mythology of Nazi Germany, Islamic texts in the case of ISIS have merely been mobilized to re/create a rosy and simplistic vision of Islam's golden days to which adherents are expected to believe in and work to resurrect.
To say that ISIS is "very Islamic" strikes me as an odd declaration, like saying "the Ku Klux Klan is very Christian". The KKK uses Christian symbolism and spokesmen often talk about restoring the United States to its original (white) destiny. I'm sure many Christians would take issue with such a comparison. They may even feel personally offended by such a charge - and this is exactly what I believe many Muslims feel when they are told that ISIS has an undoubtably Islamic core. No amount of lip service to how other Muslims condemn ISIS can cover over the veiled accusation that there is something inherent in Islam that makes Muslims susceptible to anti-Western sentiment, violence, or outright terrorism. Otherwise, why not raise parallel concerns over the correlation between violence and Christianity?