Medieval Romans tend to disagree and I prefer to believe rather them than you.
It's difficult when you mix improper terms like the "Greek Empire" (it was never a "Greek Empire") and consider Soviet Union as something completely different from other stages of the evolution of the Russian state with "soviets" apparently living there instead of Russians and all other ethnicities.
But ok.
Imagine a Russian family, living in Moscow under tzar Nikolai II. They were Russians.
Times change, bolsheviks won the war, there's Soviet Union now.
This family still lives there. The state is called Soviet Union now, but they never ceased to be Russians and never replaced their identity with "Soviets".
Soviet Union fell apart, there's Russian Federation now. This family still lives in Moscow. Nothing changed, they are still Russians, as they were 50 and 100 years earlier.
They experienced switching to a successor state two times but remained who they were - Russians.
Now the Roman state.
There was a Greek family, living in Byzantion (let's play as Greeks here

) under the reign of Hadrian. They spoke Greek at home and considered themselves citizens of Byzantion (their polis).
Caracalla said, that all free people in the empire are now Roman citizens. Therefore our Greek family from Byzantion acquired Roman citizenship and became Roman citizens. They still spoke Greek at home.
Hundred years later Constantine I arrived and renamed their city to Constantinopolis. Family members had the Roman citizenship for 100 years by now. Therefore they call themselves Romans now, but speak Greek at home (though maybe they learned a bit of Latin, why not? It was probably helpful sometimes).
Another hundred years passed. Theodosius died, Arcadius ascends to the throne. They still live in Constantinopolis. They consider themselves Romans, as everyone else around. They speak Greek at home.
More than 100 years passed again. Justinian builds Hagia Sophia. They still live in Constantinopolis. They consider themselves Romans. They speak Greek.
500 years passed. They still live in Constantinopolis. They still consider themselves Romans and speak Greek.
204 years passed, barbarians captured the city. They still consider themselves Romans and don't want to call themselves "Latins" or "Genoese" or whatever else. They speak Greek.
Few decades passed, Constantinopolis was reclaimed by Romans. They still live there and still speak Greek, while still considering themselves Romans, because few decades wasn't long enough to switch culture for them.
200 years of considering themselves Romans passed and finally Mehmed the barbarian captured the city. This time there was no Romans anymore to reclaim it. They still considered themselves Romans and still spoke Greek.
What happened next? Two possibilities:
They still considered themselves Romans and kept Greek language (there was a lot of Greeks speaking Greek in the Ottoman Empire)
They adopted Ottoman culture and after some time became Turks.
Before that at no point the chain was broken. Since acquiring the Roman citizenship and becoming Romans - they did not lose it in any way. The state continued to exist and wasn't replaced by something foreign. That happened only in 1453 and that's where succession breaks.
No, the historic moment for the starting of the process that woud lead the Eastern Roman Empire to be a Roman Empire - the sole survivor of the Roman empire (when there's no "western" there is no need for "eastern" anymore). It's not that hard to understand.
If you have a cake, cut it in half and eat half of it - the other half is still a cake, it won't become spaghetti just because of it.
And did you miss the so called "Empire of Nicaea" which was the direct continuation of the Roman Empire? It wasn't something completely new. It's not that people living there considered themselves Romans and the next day suddenly said to themselves "oh, okay, we're not Romans anymore, we're Nicaeans now". They were still citizens of the same state, just the imperial court was temporarily located in Nicaea now and not in Constantinopolis.