To my mind, calling them Greek is a bit like calling people in, say, Belize "English". If I brought a vase from Belize and said "I brought you an English vase", would you not find that odd?
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.
> To my mind, calling them Greek is a bit like calling people in, say, Belize "English".
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.
Byzantine = specific, inaccurate. Romans = accurate, nonspecific. Greek = a bit of both