“The violence was occurring long before we even got there”
Well, no in fact it wasn’t actually. That is not to say we caused that violence, because we did not. But the security lapse between the end of Saddam’s reign and reformation of new security measures was used as an opportunity for mobs/militias to exploit the security situation for their own ends, and some of the corrupt politicians have also been part of that.
It is not a fiction that Saddam built the bulk of his political machine with Sunnis or that that machine was particularly harsh on the Shia majority. (Of course it is also true that long before the Iran-Iraq war and even before the rise of Khomeini, his group was trying to organize inside Iraq) It is fiction that the majority of average everyday apolitical Iraqis, Sunni and Shia, hated each other as much as they both hated Saddam.
“Culturally” there is little difference between them, besides some different tenants of Islam that each places priorities on over others. Long before Saddam was ever in power, the Sunni and Shia shared Iraq. From time to time their religious differences have been exploited by those seeking the mantle of leadership. In the daily life of most Iraqis, until very recently, those differences did not challenge their living among each other.
But the Kurds are a different people altogether. They have existed as a distinct people in Mesopotemia since long before Islam and genetically (ethnically) they are distinct from “Arabs” or Persians, or Syrians or anyone else. They are the largest body of non-Turks in Turkey and make sizable minorities in Iran and Syria. The Sunnis and the Shia in Iraq have some religious differences, but outside of those differences they are more one people, by ancestry and common history and culture than not. The Shia in Iraq are Arab, like the Sunnis, not Persian as in Iran.