The world is a much more complex place than it was in the mid 1770s and 80s. There are compelling competing interests in the region of Turkey, Syria, Iraq, Iran, and the area in which the Kird's now live.
At the present time we have a working relationship with Iraq. It seems advantageous to pursue keeping that relationship. At the same time, we have had a working relationship with the Kurds, and we want to keep that relationship, and more importantly them safe too.
Going in and offending Turkey and Iraq is not wise > IMO. If we can maintain a relationship with Iraq and return to the time when Iraq allowed pretty good autonomy to the Kurds, that seems like the best outcome > again IMO.
Joining the Kurds would most certainly alienate Turkey. It has a claim to self-determination, running their nation as they see fit. It would also alienate Iraq which also has a right to self-determination, running their nation as they see fit.
It makes sense to me to defend the Kurds from abuse at the hand of Turkey and Iraq diplomatically.
Leave the status quo as is, and do not drive Iraq and Turkey further into the arms of Iran.
I respect the Kurd people. I do not want to see them abused. If push came to shove and it looked like Turky and Iraq were going to attack them, seek to destroy them, we might have to escalate our support for the Kurds.
That is how I could play it for now, and hope things returned to a stable status quo which would work for all three parties, the Kurds, Turkey, and Iraq.
And they each have their issues, so it’s not simple situation. IMHO which is far from scholarly, the more hands off the US is the better, while being sort of nice to everyone. Except less nice to Turkey. Maybe some tsk tsk sounds too. Firmly taking any sides here would be a big mistake, again IMHO.