since Byzantium in the end really just is Greece with an Emperor on the throne
I disagree. The Empire had plenty of non-Greek emperors. There were Slavic and Armenian emperors and more than once did they take the throne. The Empire was hardly an ethnocentric state as nationalism didn't exist that time. The cultural life of the Empire however, was most certainly Greek but whether that cultural life is more akin to ancient Greece or to today's Greece is a question with a hard answer.