The principal problem with this idea is the question of 'who are the Kurds?' Those people who self-identify generally as Kurds are a very diverse group. They speak several different languages, some are referred to as 'dialects', such as Kermanji and Sorani, but which are not intelligible one with another. There are also completely separate languages such a Zaza and Gorani, which other Kurdish speakers don't recognise as Kurdish at all. They are a hugely diverse group religiously too, with some half-a-dozen faiths all having major followings. Sunnis, Shi'as, Alevis, Sufis, Yarsan, Yazidis and some Christians all claim to be Kurdish. Trying to forge all these different faiths and languages into one homogenous nation would be no easier than the nation-building efforts in Iraq. The Kurdish region of Iraq, which so many people seem to hold up as being the model, has much less diversity than any Greater Kurdistan would have. The second such a state was declared, it would plunge into in-fighting and disorder.
I'm not saying Kurds don't deserve self-determination, it's just that they don't really have a clear idea of self. I very much doubt that a Syrian Yazidi Kurd would have anything very much in common with a Khorasan Iranian Shi'a Kurd, not culture, language, religion or history.
So, before the West decides to help the Kurds with their nation-building (when has that ever ended well, btw?) they need to wait to see who claims to be leading the Kurds and where they think that nation should be situated - it could be anywhere from the Med to the borders of Afghanistan.