If Romans were Greek (the inhabitants of Latium were not Romans until after the citizneship law of AD 212), and Greeks of Hellas became Romans through conquest, then the Greeks of Magna Grecia were Romans, and that they now speak a Latin tongue means they were Latinized.
But the most learned of Roman historians, among who is Porcius Cato, who compiled with the greatest care the genealogies of the Italian cities, Gaius Semporonis and many others, say they are Greeks, part of those who once dwelt in Achaia, and migrated many generations before the Trojan war." (Dionysius of Halicarnassus, Roman Antiquities, I, XI)
Certainly your kith and kin in Greece and Turkey and Syria call themselves Romans, or Romaoi, as did and do their Muslim opressors, who fancied themselves as Sultans of Rum, or Rome if you will.
You are exactly right. There never was a "Byzantine Empire" -- this term was invented in recent centuries by a French historian, I believe. Orthodox have always considered themselves to be Roman, as you say -- even to this day, especially in the Middle East. It is the origin of the term "Rum Orthodox."
Fr. John Romanides, in his "Franks, Romans, and Feudalism", goes out of his way to make this point, and he paints the struggles from the 8th to 13th centuries not as ones betweeen East and West, but between Christian Romans (East and West) and the new Frankish political entity, which understood the power of exploiting the Church to achieve its ends.
One can agree or disagree with his analysis, but one will never look at that time period again if one reads this work.