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The Levant was largely hellenized since ancient times. There's a reason the New Testament was written in Koine Greek rather than Aramaic or Hebrew, as it largely replaced Aramaic as the lingua franca. Obviously actual Greeks never became predominant there outside of some of the Greek-founded cities, but their culture became the ruling one. Also the forced resettlement of population groups was fairly common in the ancient Middle-East.
Either way, the Roman Empire also had some emperors who weren't of pure Latin ancestry, so it's really no surprise it also happened to their eastern iteration. Also the Byzantine Greeks saw themselves more of a continuation of Rome rather than Ancient Greece, I don't think they actually cared much about this sort of thing.
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