When Greek rebellions finally prevailed in the Ottoman Empire, why did they not choose to identify as Romans instead of Greeks? Being that the last state they would have been lived was the Byzantines, and they would have identified as Romans and not Greeks then, and that the area had for almost 2 millennia had been Roman, not Greek, it seems a little strange to me that the modern Greeks would look so far back to the original Hellenic Greeks as their parent civilization instead of Rome. Was this just about geography? Because that had never stopped people from claiming they were Roman before. Or because of the still-recent rebirth of Greek art and writing (which would’ve also been accompanied by Roman art too but I guess that’s besides the point)?
Some Greeks still did cling to the Byzantine rather than the Hellenic legacy, including the Church and (to some extent) the early revolutionary Rigas Velestinlis—I went a bit more into these alternative proposals for the Greek national heritage in this answer a few months ago. These are the ones who wrote down their views and thoughts about the future of the Greek national heritage. We can't tell what the average peasant might have thought, but it seems that neo-Hellenism was a phenomenon largely restricted to an intellectual class; the average Greek probably inclined more towards the Byzantine model.
The standard academic view has generally been to regard Greek enthusiasm for the classical past as something of an importation from European discourse. David Brewer points out that the 1750s began a period of translation of a large number of Enlightenment works into Greek—including political writings calling for the liberation of Greece, as well as works about and set in ancient Greece. Enlightenment values thus informed a number of influential revolutionaries; and these values were not always conducive to a close link with the Byzantine past.
For example, quite a few early Greek pamphlets are tinged with a stern anticlericalism that might cause issues for anyone hoping for a return to the Byzantine past. The Ellinikí Nomarchía, an exposition of the revolutionary cause written in 1806, has this to say about the matter:
A hundred thousand, perhaps more, monks [ ... ] live idle and feed on the sweat of the unfortunate and poor Greeks. So many hundred monasteries, which are to be found everywhere, are such wounds to the motherland, for without being in the least beneficial, they eat its produce,and harbour the wolves, [who] seize and tear to pieces the innocent and peaceful sheep of the flock of Christ. See, O Greeks, my beloved brethren,the present wretchedness and fearful state of the Greek priesthood, and the primary factor which delays the freedom of Greece.
In addition to translation, new educational institutions opened up in the Western Ottoman Empire with emphases on classical Greek culture. These included colleges in Bucharest, Iasi, Chios, and Smyrna, among other places, and provided a medium for disseminating ideas about classical Greek heritage to Greeks, as well as to other subject populations within the Empire. At the same time, societies like the Philomousos Etaira in Vienna helped fund young Greeks' education by sponsoring studies abroad (in European countries where the Hellenic heritage was stressed) as well as archaeological excavations. In this way, the intellectual class was able to impart their ideas about the Hellenic past and its importance for the present to new generations of scholars.
After the revolution, the Hellenic view of the past was slowly promulgated through all classes of Greek society through time, education, and a lot of government support—but the exact mechanics of that shift are a bit beyond my specific knowledge.
SOURCES
Richard Clogg, The Movement for Greek Independence, 1770-1821: A Collection of Documents (Macmillan, 1976). Contains translated excerpts from the Ellinikí Nomarchía (quoted above) as well as a useful introduction.
David Brewer, The Flame of Freedom: The Greek War of Independence, 1821-1833 (London: John Murray, 2001). Especially chap. 2, "Resentment and Regeneration," 8-16.
Cyril Mango, "Byzantinism and Romantic Hellenism," Journal of the Warburg and Courtauld Institutes 28 (1965): 29-43, doi:10.2307/750662. An interesting if dated examination of Greece's sometimes-contentious relationship with the Byzantine past.