Perhaps atheism isn't an ideology—which would explain why the New Atheists instead peddled a bunch of basic neocon fake "clash of civilisations" crap that seemed to have nothing much to do with Christianity or its little brother, secular humanism, when they were advocating for the intervention in Iraq.
The claim that the absence of religion is non-ideological is not too different from a claim that the presence of religion is non-ideological. And to a much greater extent than we are told, it is—the "sectarian" struggle in the Middle East is also about contending imperial constellations, Russia-Iran-Syria vs US-Turkey-Israel-Saudi-Qatar.
We've all read about how Iran and Saudi Arabia are unhinged, authoritarian religious regimes, but they are not so irrational as to be unable to form political alliances that endure for decades with decidedly non-Islamic powers, or to play elaborate games manipulating energy markets. Perhaps there's more going on here?
One of the main things they're fighting over is control of the fuel supply to Europe, the other bit is that no one is willing to tolerate the continued energy dependence on Saudi.