There is no such thing as Greece before 1821. So they would have to travel back in time.
Byzantine = specific, inaccurate. Romans = accurate, nonspecific. Greek = a bit of both
Just because someone speaks a related language (and I'm pretty sure the Greek of Constantinople was different from the Greek of Athens at the time), doesn't mean that they are the same people. The Byzantines had hundreds of years as a distinct culture from the Greek islands and peninsula, with a major Roman influence.
Those situations are nothing alike. The Byzantines lived in Greece. Byzantium was founded by Greeks. Others in Europe called them "the Greeks". They were the genuine continuation of Hellenic culture for over 10 centuries.
It's either that or Greeks ceased to exist between Roman conquest and Ottoman independence - at which point they were ruled by a German and, presumably by your logic, were actually Turks anyway, not Greeks.
If you know much about Hellenic history, you know it's been a culture in flux since prehistory. I'd assert there has never been a group that you would call "true Greeks"... except maybe the Graecoi - Hellenic colonists in Italy. Even the post-Classical period of pan-Hellenism was driven and ruled by Macedonians, who a century prior were not considered Hellenes.
I mean, define 'Greek'. Byzantium was a Greek city before the Romans got there, Greek was always its major language, and so on. It's not within modern Greece, granted, but nor are a lot of classical Greek cities.
Also, the culture of Athens or Sparta or Crete or any of the other places that would have called themselves, or at least accepted the term, Greeks (well, Hellenes) was quite different from the culture of Constantinople, at least, again, by the time the city came to be known by that name.