The debate about what Byzantium was and whether it was Greek has been going on for many years and it will no doubt continue.
The facts are there to be looked at. The Byzantines spoke Greek, were of the Greek Orthodox faith ( they still are ) and the emperors that ruled the Empire for the last 400 years of Byzantium's existence were all from the Greek mainland Dutchies. So the Paleologos dynasty which dominated the scene for 230 years before the collapse were the Despots of Mystra in the Peloponnese.
The question should not be whether Byzantium was Greek but whether modern Greece is what survives of the Byzantine Empire? The culture, the language and much of the laws still have a Byzantine origin. Modern Greece is still the last part of what was the Byzantine Empire. All one has to do is travel through out Greece to see the Churches and the monasteries that dominate the country.