Something I think people miss is that the Byzantine Empire was, for much of it's existence as something different from the Roman Empire, an Anatolian thing. It's geography looks more like the Hittite Empire or the modern Turkish Republic, and mainland Greece was not the bulwark of the Empire - it was a periphery that had to be retaken militarily and then reintegrated. When you genetically test a bunch of people who would identify as "Romans" between 800-1150 you're going to test a bunch of people from Western Anatolia and the islands. The picture would be different if you looked only at the remnants of the Empire in the Balkans in, say, 1320.