The basic assumption seems to be that the Bible gives an accurate history of the past, including a reliable chronology. From that assumption springs the idea that the 10,000 years date must be wrong. The fact that the site is near where some think Mount Ararat was located, and that the site includes many depictions of animals is too good a set of circumstances for someone who is already inclined to take the Bible account seriously to miss making a connection. If the Bible is correct that Noah existed and saved animals on an ark then the next thing is to say that he could also have built the Göbelki Tepe site. I think Watchtower has at times suggested that Noah could have employed workers to help him and his family build the ark. Perhaps Stafford is reasoning that, if Noah could hire help for the ark, then he could have hired help to build Göbekli Tepe too. What I am not sure about, and maybe I just missed it in the video, is what function does Stafford think this site performed, and why wasn’t it mentioned in the Bible? Was it a kind of holding bay for the animals before they went on the ark? Why was that necessary?
There is another JW apologist who has attempted to link Noah’s flood with the ‘younger dryas cataclysm’ that has piqued the interest of some in recent years. I forget his name or the name lf his book right now.