Yes, the Kurds deserve support for an independent homeland. But.
I concede that the idea that Kurdish independence harms the fight against ISIL is incorrect. It rests on the dubious argument that Iraq will hate the Kurds so much that they will divert force from the fight against ISIL to take on the Kurds. That's hardly a ringing endorsement of Kurds remaining within Iraq, eh?
But, there are reasons to be cautious nonetheless.
We can hardly support Kurdish separatism in Turkey as long as Turkey is a NATO ally. Bad form that would be. Sure, Turkey isn't being a very good ally lately (although I'd go back to 2003 for the start of their bad policy trend). But as long as they are our ally, we have an obligation not to work for splitting their country up.
I'd have no problem supporting independence for Kurds in Syria and Iran. That hurts our enemies.
Yet while I have complete sympathy for the Kurds of Iraq wanting an independent state, unless America can convince Baghdad that an independent Kurdistan split from Iraq is a good idea for Iraq, too, an independent Kurdistan cannot survive surrounded by hostile states (Iran, Iraq, Syria, and Turkey).
If Iraq accepts an independent Kurdistan and the formal loss of Iraqi territory, then Kurdistan has an outlet to the sea and can trade with the world and thrive. If not? Kurdistan would be doomed. Then it would be better for the Kurds of Iraq to negotiate more autonomy that provides practical independence from Iraq without legal independence.
Even then I have worries about Iraq losing the Kurds who provide a measure of anti-Iran muscle within Iraq which reassures Sunni Arabs that they can balance the Iraqi Shia majority (only some of whom are pro-Iran, to be clear).
Although the Sunni Arab embrace of ISIL--regardless of how quickly they regretted that move to oppose the Shia-dominated government--even while the Kurds were within Iraq lessens this consideration over time.
I hope America works with the Iraqis and Kurds to gain a measure of mutual trust for an amicable split such as Czechoslovakia achieved. But I hope that we can make enough progress to stay the Kurds from unilaterally going on a suicide mission.
Yet with Iraqi Shia militias (popular mobilization forces) with less to do as ISIL goes down to defeat within Iraq, I worry that pro-Iran elements might trigger a fight with the Kurds--like near Kirkuk.*
*Although I'd like to point out that the "tank" pictured in that article and captioned as retaking Kurdish positions is odd given that the "tank" is a self-propelled artillery place and unlikely to be taking a frontline role.
UPDATE: I don't think it is fair to say America "stabbed the Kurds in the front" (as opposed to in the back) by opposing their independence bid. Is it really pro-Kurdish to support formal independence when it would just lead to a war against the new state by Iraq, Iran, and Turkey? Which would be bad for the Kurds as well as bad for our broader interests in the region?