Why does modern Greek historiography not view the Byzantine Empire as a Greek Empire?

You're not wrong that Greek nationalism tends to emphasize ancient Greece over the Byzantine period, but it would be a bit of an exaggeration to claim that the Byzantine Empire plays no role in modern Greek national historiography.

That said—when the Greek state achieved its independence in [1829-1832, depending on who you ask], nationalists certainly tended to downplay the role of the Byzantine empire in Greek history. To some extent, this was due to the "marketing power" of Classical Greece—the European philhellenes who had sponsored and participated in the war with the Ottoman Empire were inspired by Themistocles and Leonidas, not Justinian and the Komnenoi—but it's also important to note that many of the most important figures in the revolution were themselves educated and sometimes foreign-raised Greeks who subscribed to the same valorization of the Classical past.

The other issue is that the Byzantine Empire was far more multinational than Classical Greece, and was defined (with some exceptions) more by its Christianity than its Greekness—so while Greeks intent on a national state could evoke the glories of Athens and Sparta with no real competition, the Byzantines could be partially claimed by Serbs, Bulgarians, and other Balkan subjects the revolutionaries never intended to include in the Greek state. While this led nationalists to downplay the Byzantines, it also made the Byzantine Empire the model for those who valued religion rather than ethnicity as the most important identity—a group including the early revolutionary Rigas Velestinlis as well as the Church. (I talk a bit about the Church's distrust of philhellenism here; and in fact, continuing power struggles between Church and state after 1830 may have contributed to the low stature of Byzantine studies among nationalist historiographies in the decades immediately following independence.)

1830 was significant for another reason, however: this is when the Austrian scholar Jakob Philipp Fallmerayer first proposed that the ancient Greeks had been entirely replaced through successive waves of Slavic occupation, and that contemporary Greeks were Slavs in all but language. Suddenly, demonstrating the continuity of Greekness past the battle of Chaeronea (the traditional endpoint of Greek independence before the revolution) became really important! Fallmerayer's thesis occasioned a great deal of outrage among Greek intellectuals, among whom was Konstantinos Paparrigopoulos, the most prominent Greek historian of the mid-nineteenth century. Paparrigopoulos's History of the Hellenic Nation [Στορία του Ελληνικού Έθνους] (1860-1874), which would remain the dominant text in Greek history education through the early twentieth century, proposed a continuous history of the Greeks in three periods: the ancient, medieval, and modern. Paparrigopoulos stressed the spiritual and cultural continuity of the Greek people over any racial continuity, but he—along with his colleague Spyridon Zambelios—has been credited with spearheading the rehabilitation of the Byzantines (as "Helleno-Christian," ελληνοχριστιανικός) in Greek academia. By 1914, the state itself recognized the trend by decreeing the establishment of a Byzantine Museum.

This mid-nineteenth-century rehabilitation of the Byzantine empire dovetailed quite nicely with some important national and international trends. Politically, it helped irredentists draw on the Byzantine empire for their territorial claims of a greater Greece (the so-called Megali Idea). Aesthetically, as European tastes began to shift from the philhellenism of the early Romantics to the Gothic Revival, the desire to keep up with fashions may have contributed to the growth of Greek medievalism at home.

Of course, the Megali Idea ended up being impracticable on a large scale (as demonstrated by the attempted annexation of Cyprus in 1974) and the Gothic Revival gave way to new artistic and architectural fascinations. Greece's attitude towards its Byzantine heritage, which never really rivaled that towards the Classical period, is probably a bit cooler now than it was in Paparrigopoulos's day. I don't want to comment too much on contemporary Greece, since the twenty-year rule is a thing, but take a look at this clip from the opening ceremony of the 2004 Athens Olympics, and you tell me whether the Byzantine Empire is entirely ignored!