this is what iraq didn't have. i mean, iraq had and has kurds, but the kurds cannot govern iraq. iraq obviously has a very long history but iranian groups have only ever existed in iraq as occupiers. arabs are not indigenous to southern iraq, but they've been there a long time.
the origins of the kurds are not clear. they clearly speak an iranian language, and are broadly pretty europoid and white-skinned, which together indicates that they must have entered the region somewhere around the collapse of the neo-assyrian empire, but that process is itself not told well by history and lost in the poetry of herodotus.
the word kurd is not an iranian word but an arab word, akrad, that translates as "refugee". the kurds have lost their self-designation and adopted the word of their oppressors, but it also masks the fact that the kurds are not exactly a tribe or an ethnicity but a group of people defined by how and where they live, rather than the language they speak, or the tribes they derive ancestry from. there was a non-iranian group of unclear linguistic or ethnic origin called gorduenes living outside of the zagros mountains not very far away, but essentially all academics consider that a false cognate (like goth with guti). the arabic origin of the word kurd is the academic and scholarly position.
i've generally adopted occam's razor about this. the history of iraq, as written by the arabs and muslims, does not explain what happened to the iranians or greeks of iraq. the quick history lesson is that the persians took over iraq when babylon fell (about 550 bce), were defeated by alexander and then came back a few centuries later. there were longstanding wars between rome and persia in mesopotamia and armenia and syria, but iraq was usually a part of persia, or parthia, from the return of the parthians to the islamic conquest in the 7th century. overall, that's 1200 years of persian rule, minus a short period of hellenic hegemony. where did they go? the muslims don't tell us.
i think the answer is the most obvious one - they fled southern iraq, became refugees in the mountains and have been the kurds ever since. the arabs labelled them akrad at some point and the word stuck.
now, it's worth pointing out that the kurds, once they do appear in the muslim sources, become an important force in the region. islamic mythology talks of a "golden age" that was anything but; it was a period of relative scientific ignorance that should be described as a dark age, similar to that which occurred in western europe. however, the eastern roman empire was older, richer and had deeper infrastructure, so the dark age was less pronounced. notably, the arabs took over all of the libraries. that was a coincidence that had to do with roman administrators building libraries in warmer, drier climates where the books wouldn't decompose. like, they wrote books on leaves and sheepskin; paper was invented in china and doesn't get to rome until after it collapses. you just couldn't have books in europe because it was too cold and damp. you had to have them in egypt or syria, where they could last. so, the dark forces of religion dismantled both sides of the empire, and the arabs, as the barbarians they were, did everything they could to generate a dark age in the eastern empire, but the east still had the books and the west didn't. the books didn't really come back to the west until the fall of constantinople, when they were brought back to rome by an elite fleeing the lost city. by this time, the byzantines certainly had paper, but they hoarded knowledge from the outside world. the late roman empire maintained technological supremacy for decades by going so far as encrypting basic scientific literature, to prevent the barbarians from learning it. they did not believe in progress, science or an open society, either.
but the arab dark age in the former eastern empire that followed it's collapse due to the rise of religion, and that islamic mythology calls a golden age, was offset partly by what could be called a neo-babylonian renaissance in parts of iraq that was made possible by the end of centuries of war between rome and persia. and who is responsible for this? the answer is kurds. you can go through the list of so-called golden age scholars, and you'll note that virtually all of them are kurds, that they spoke iranian even when they wrote in arabic, and that most of them had issues with the muslim religious authorities (similar to the conflicts that renaissance italians had with the popes in italy) due to heresy, syncretism (shiiite mysticism) or flat out paganism. there remain a number of minority religions in iraq.
what i'm trying to explain is that if the kurds were the descendants of the iraqi persians that had been there since cyrus then they never really left. they were still there and still running the country when the dust settles after the muslim massacres. but they were always occupiers. they were occupiers at the fall of babylon, occupiers during the classical period, occupiers during the islamic dark age, when the babylonians became arabs, and occupiers during the babylonian renaissance that followed.
the kurds can govern iran, because they are iranians. they will be expected to behave like iranians, and not hoard power. but, they can get things going in iran, and could not do that in iraq, where they would be constantly seen as outsiders.
