I don't think you know what the Eastern Roman Empire was or what Romans were. You also think 500 years of history are monolithic. Yes, Romans of 500 AD were culturally distinct from Romans 100 BC, including who could be Roman citizen and then even Emperor. It is still a continuation. That equally applied to all parts of the Roman Empire. Rome also never established hereditary dynasties so many people could gain Roman citizenship, even join the aristocracy and within centuries became Roman Emperors without having been born anywhere close to Rome.

Romans used Greek as the sophisticated language in the higher classes already in the republic. As such it was favored long before the "Fall of Rome" by Romans.

The division into West and East Rome was an administrative organization of the Roman Empire. As such East Rome was an uninterrupted continuation of the Roman Empire within its eastern administrative borders.

And that the Pope, who usurped imperial authorities or Germanic dynasties who grabbed territory did not accept Roman authority is somehow expected. Though even that is a simplification because apparently the West Romans did no see their empire fall when historians claim it did. They even complained about Justinians reconquest a as a needless civil war between Romans aka did not consider themselves under foreign rule.