I have not seen any DNA on the people of Gobeki Tepi. We don’t have any reports of significant movement of peoples into anatolia between the time of Gobeki Tepi and the movement of anatolian farmers out of the area starting roughly 5000-6000 bc. so it may be safe to say that they are the same people.
what is know is that these anatolian farmer crossed into greece where there were western hunter gatherers. Over the next millenium or so they spread out throughout europe and mixed with resident hunter gatherers including Greece.
About 3000 bc the various groups of the pontic steppe or what is now ukraine and parts of russia moved west into europe and balkans.
All of this happened before the pharoahs built their pyramids. The old kingdom didn’t start up until about 2600 bc.
The Old Kingdom didn’t start up until about 2600 bc.
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The Archaic Period was before that around 3600 BC - however all the dates are based on various lists of Pharaohs, and one or more of these lists are known forgeries, yet they are still used.
As for the names in the lists - many of the early kings are unknown, others are unknown unknowns (part of their cartouches are missing), others have the same names, others have one name at one point and a different one at a point, others omitted for political reasons. In short, it is a confusing mess, but still standard Egyptology uses them for dates, which are likely completely wrong.
Another example is an optically stimulated luminescence date was taken from the Pyramid of Menkaure, only to turn out to be some hundreds of years older that the others, yet this was the newest one. More confusing still is the entire Gaza complex was built as one geometrical/mathematical unit, not separate structures built over time.
So 2600BC is far from a firm date, rather it is just a generally accepted date that keeps the current narrative intact - a narrative going back to Herodotus with just the edges tweaked, and the money from tourism pouring into Egyptian coffers.
When tracing the movements of peoples in that area its would be best to ignore any Egyptian dates as that will just make a mess later on.