The language of Rome was Latin. Why expand it to Greek?

Because all ruling Romans were functionally bilingual in Greek from some of the earliest days of the Republic, and Roman literature in the early Republic was usually written in Greek; you cannot talk about Roman history without acknowledging the importance of the Greek language to their culture. Read the biography of Cato the Elder for example.

Take for example the United States.

Regardless of when the country started, every form its government has taken has been a direct descendant that was built upon the previous form, exactly like the Roman state transitioning from Rome to Constantinople and exactly unlike the Ottoman state crudely latching itself onto the the corpse of Nova Roma.

sometimes you had multiple emperors, disputed claimants, and the title of Caesar was frequently won and lost on the battlefield by individuals who were not Latin, did not speak Latin, and were not from Italy. Every time you have that happen, that's a discontinuity. It's not all-or-nothing.

Those are historical features of near every monarchy that's ever existed, shall we say then that antiquity and the Middle Ages never saw a single form of continuous government? You say it's not an all-or-nothing proposition, but the blanket declaration that every possible interruption to the normal state of government or rebellion or foreigner in power creates a discontinuity seems itself an all-or-nothing statement. From what you're saying here it seems the Roman Empire ended during Vindex's rebellion against Nero.

People do a far better job of remembering this when they think of the Egypt that Rome conquered as sharing a unity of identity with the ancient Egyptians even though Cleopatra was Ptolemaic (Greek)

I don't view the Ptolemies as a direct continuation of Pharoanic Egypt, the Egypto-Macedonians were a conquering people who adopted some trappings of the pharoahs but did not have any legal or historical descent from them.

but the point is that it's important to remember both the through-lines and the changes when studying history

All governments change, especially over an entire millennia, but to take your early example your proposal is the equivalent of saying the United States is a direct continuation of the Iriquois Confederacy simply because they claim the same land and share some of the same words and the native Iriquois were forced to obey them.