The claim to establish the basis for the separatist movement in Xinjiang, which was picked up and promoted by the State Department is that of East Turkestan, which is essentially separating Xinjiang from China and creating an Uyghur/Turkic ethnostate (one that makes no sense because Uyghurs are also not solely Muslim, they don't form one homogenous group, and the province has long been the home of many different ethnicities. Uyghurs themselves are primarily descended from the Qigour/Ouigour people, who are not Turkic whatsoever).

The basis of the East Turkestan claim is advanced by the likes of the World Uyghur Congress, who gives themselves an authoritative name but is really just an arm of the NED with ties to other far-right organisations that fight for the same dream. Here is a page on the WUC website clearly laying it out:

Deconstructing the claim that Uyghurs are Turks or sufficiently related to them to give a foundation to the East Turkestan claim (which is part of the broader Pan-Turkism ideology, a fascist idea) is primordial to deconstructing the whole narrative around the "genocide". But you're right that we glossed over the overall importance and context of why this matters, we assumed that everyone would already aware that the US was a proponent of the East Turkestan theory and could have made the links clearer.

The USA of course is hoping that by funding, training, and promoting a separatist group, there will be an independent Xinjiang who will sell their oil rights to them and cut China off from the belt&road land route.

Edit: note that the contentious point was about the Turkic/Turk/Turkish wording. I think all of us involved in that discussion on Twitter speak English as a second language, so perhaps some connotations on the differences between these three words escaped us during the discussion.