I'm not an expert in Greek history or culture, so I'll gladly defer to someone who is for a more nuanced answer here. But after nearly a week no such answer has emerged, and it happens that in recent months I've done a fair bit of research on this very question. It's just that my research has been for my French history podcast, and specifically an episode on the Greek War of Independence, from which this answer is largely adapted.
When the Greek War of Independence began in 1821, Greeks in the Ottoman Empire tended to refer to themselves with the word "Romioi," or Romans. The alternative was "Hellene" — the name used today in Greek for their country and people — but neither "Hellene" nor the Latin form "Greek" was commonly used by the Greeks themselves circa 1821.
That makes sense, because the Byzantine Empire was the dominant touchstone for Greek identity, not the classical polities of Athens, Sparta and Thebes. Not only had the ancient Greeks died millennia ago (compared to a few centuries since the fall of Constantinople), but they were pagans to boot. While Athens would eventually become the capital of an independent Greece, the dream in the years leading up to the rebellion was of "the romeïko," a quasi-mystical day of reckoning when Constantinople would be liberated due to some combination of God and men.
But while the Greeks themselves circa 1821 identified themselves with the Eastern Romans, there was an important group of people who had a very different image in their head of the Greeks: Western "Philhellenes" raised since childhood on the Iliad and the Odyssey, on the 300 Spartans at Thermopylae, on disputing philosophers on the streets of Athens. The Byzantines were largely ignored or dismissed. And while the Roman identity predominated in Greece itself, sometimes elite Greeks who obtained Western educations came to identify more with the classical Greek image over the medieval Roman one.
Even the languages were different! The classical Greek taught in western schools was essentially a completely different language from the living Greek spoken in Greece itself. One illustrative example comes from the experience of an Englishman, fighting alongside the Greeks in 1825, who recited a stirring passage of Homer in classical Greek to his hosts. The response was a question from one of the Greek fighters: “What language is that?”
But as the Greeks began fighting an uphill battle for independence against the vastly more powerful Ottoman Empire, Western support became increasingly crucial. Philhellenes sent money, weapons and volunteers — and in doing so, they were usually motivated by classical stories. A Danish volunteer wrote, "I had learned to admire the Greeks from my school days, and how could a man inclined to fight for freedom and justice find a better place than next to the oppressed Greeks?"
And pro-Greek propaganda published in the west capitalized on this. Brits, Frenchmen and Germans read manifestos supposedly written by Greek leaders that were loaded with references to classical battles and figures who most modern Greeks had never heard of. Here's one such manifesto, which in this case was written by an actual Greek — albeit the Europeanized Alexander Ypsilantis, who had served for years in the Russian army:
Let us recollect, brave and generous Greeks, the liberty of the classic land of Greece; the battles of Marathon and Thermopylae, let us combat upon the tombs of our ancestors who, to leave us free, fought and died. The blood of our tyrants is dear to the shades of the Theban Epaminondas, and of the Athenian Thasybulus who conquered and destroyed the Thirty Tyrants — to those of Harmodius and Aristogeiton who broke the yoke of Pisistratus — to that of Timoleon who restored liberty to Corinth and to Syracuse — above all, to those of Miltiades, Themistocles, Leonidas, and the three hundred who massacred so many times their number of the innumerable army of the barbarous Persians — the hour is come to destroy their successors, more barbarous and still more detestable. Let us do this or perish.
These references would have been gibberish to most of the warlords doing the actual fighting in Greece. But it was catnip for westerners.
The case of Ypsilantis underscores the importance of westernized Greeks themselves — and not merely classicized foreigners — to this shift in Greek self-perception. The Smyrna-born Adamantios Korais moved to France and became a notable savant there, and in his works he prioritized Greece's ancient heritage over its medieval Orthodox Christianity. Some significant Greek revolutionary leaders had western educations and leanings — so different from the outlook of the bulk of their countrymen. But in the cultural tug-of-war, these educated Greeks who emphasized the "Hellene" over the "Roman" had an ace up their sleeve: as the war went on, westerners became ever more vital to the revolution's prospects. This climaxed when a joint British-French-Russian fleet effectively won the war at the 1827 Battle of Navarino, and a Bavarian prince was crowned as the first King of Greece in 1832.
As historian Mark Mazower wrote:
Thanks above all to the educated young revolutionaries who drafted the hundreds of decrees, proclamations, and instructions that were the output of the wartime provisional governments, "Hellene" triumphed. As it lost its originally exclusive reference to the distant pagan past and came to refer to their descendants, the modern Greeks, so the uprising itself ceased to be seen as the fulfillment of the romeïko and came to be viewed as the resurrection of an ancient nation. It was the war itself that popularized this new vision of a political community, one based less on a shared allegiance to the patriarch of the empire and more upon genealogy. "You Greeks have something grand in your heads," the all-powerful Pasha of Jannina is said to have said to Christians in his service. "You no longer baptize your children Ionannis, Petros and Kostas, but Leonidas, Themistocles and Aristides! You must be cooking up something."
In a broader sense, the ways in which the religious and imperial goal of the romeïko gave way to the ethnic and nationalist ideal of Greece is the story of the Greek Revolution — and, in some ways, the story of the 19th Century.
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