Much more can be written about this. This is not to discourage further discussion, data, or debate.

I find an interesting discussion under "Is the modern state Greece truly a successor to Ancient Greek culture?" . /u/Daeres has the main answer at the top. (If you want to digress, some replies in that subthread touch on China and Persia, and /u/[deleted] touching on the entire notion of "ethnicity".)

I found some posts addressing modern terminology for the people (demonyms), Roman versus Ionian versus Hellene versus Greek, and also the megali idea, but I don't see specific mention of the "Byzantine Empire" as accepted / rejected as a predecessor state (as opposed to basing it on culture, language, religion, et cetera). I'll mention them anyway:

• 'When/how did a Greek identity emerge from the Byzantine/Ottoman Empires? Was there any notion of recreating a "Roman" state when Greece became independent?' , by /u/yodatsracist
• "When did Eastern Romans stop calling themselves Romans?" , several comments by /u/gorat