I think Greek nationalism simultaneous rejected the “Rhomaioi” (Roman) identity Greeks had under the Ottoman Empire while wanting to recover Constantinople as the grand capital of the Greek people, from Trabzon to Crete, from Thessaloniki to Cyprus. Just like the Israeli was meant to be a “New Jew” (the Sabra), free in his own land and no longer subservient to foreign rulers, Greek nationalism sought to create a similar shift away from a subservient identity under the Ottomans (who generally called Greeks “Rum”, Romans) and towards an assertively independent identity. It did not completely reject Byzantium, but it tend to reject the name Rhomaioi. In a sense, it’s both/and rather than either/or. Greek nationalists frequently called back to the Ancient Greeks (with the term “Hellene”, with the language revival, the rebuilding of Athens) but they also called back to the glory of the Byzantium Empire at its height and had serious designs, at various points but especially 1912-1922, on taking Istanbul. This desire for Constantinople wasn’t just a late thing, however, and as early as 1844 I have found a Greek Prime Minister talking about:

There are two great centres of Hellenism. Athens is the capital of the Kingdom. Constantinople is the great capital, the dream and hope of all Greeks.

For more, see my older posts:

• Was there ever a serious effort to retake Constantinople after 1453 by the West?

• Were there ever talks of a crusade/liberation of Constantinople after it's fall to the Ottoman Turks?

For the Roman/Hellene distinction, I have a little bit here:

• 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?