The Atlantic article does not make any factually incorrect statements, as far as I can tell. But I think we can benefit from distinguishing between what ISIS says about their military goals with what they say about their religious goals.
Basically, ISIS's sole military goal is to sustain and grow an Islamic caliphate. They suspect that this will bring about the final world war that ends the world, which is in a vague sense an end goal of Islam -- it is at the Hour of the final judgment that people will be resurrected and really get to go to Heaven. But it is not their choice how soon the Hour will happen, because in Islam, it is a basic rule that no human is privy to that information:
People question thee concerning the Hour. Say, "Knowledge thereof lies only with God. And what will apprise thee, perhaps the Hour is nigh." (Quran 33.63)
Do [unbelievers] feel secure from the coming of an overwhelming punishment from God upon them? Or from the coming of the Hour suddenly, while they are unaware? (Quran 12.107)
(Yes, both these verses imply that you should behave like the world is about to end. According to the Study Quran, this is meant to serve as a kind of memento mori.)
Furthermore, there are a number of signs of the Hour which ISIS can neither create, nor would they find them particularly desirable:
The Prophet (pbuh) said: The Last Hour will not come before there come forth thirty Dajjals [false prophets], everyone presuming himself that he is an apostle of Allah. (Abu Daud Book 37, Number 4319)
“Narrated Anas: I will narrate to you a Hadith I heard from Allah’s Apostle and none other than I will tell you of it. I heard Allah’s Apostle saying, ‘From among the portents of the Hour are the following: Religious knowledge will be taken away; General ignorance (in religious matters) will increase; illegal sexual intercourse will prevail: drinking of alcoholic drinks will prevail. men will decrease in number, and women will increase in number, so much so that fifty women will be looked after by one man.'” (Bukhari Volume 7, Book 62, Number 158)
ISIS has never claimed that they will cause the end of the world, first because they cannot fulfill all the signs themselves, and second because only God is allowed to know when the end of the world will actually happen.
However, the Islamic State does see themselves as a divinely ordained caliphate that is prophesied in these traditions about the end of days. This is the meaning of the title of their propaganda magazine Dabiq, named after the town (which Syrian rebels captured from ISIS in September 2015) where the final battle is prophesied to occur, and in a few articles within it, notably one about "The Revival of Slavery Before the Hour".
In this sense ISIS fighters do see themselves as the fulfillment of divine prophecies on the grand timeline of Islamic eschatology, while at the same time being the saviors of the faith in the context of real Islamic history. An award-winning British journalist, Martin Chulov, interviewed five fighters separately and found these similarities:
Their motivations are similar, but in some cases they are diverse and contradictory. All of them draw at least some inspiration from the prophecy of an epochal confrontation in Dabiq; they see themselves as underdogs, fired by a sense of divine mission. [...]
All of these men believed that by travelling to fight for the caliphate, they were standard-bearers of their faith. They also felt sure they were acting to restore Islam to its lost glories – and had a sense of privilege and pride that their generation was the one that had been chosen to right the wrongs of the past. [...]
But they also had myriad other reasons for joining the terror group that had little to do with their understanding of Islamic scripture or any sense of holy war. Some saw themselves as victims of oppression, others as sons of dispossessed families. Another thought of himself as a cultural warrior, not a holy warrior: he argued that joining the jihad was an entirely practical obligation, necessary to restore the caliphate and bring on the prophecy of the end times.
