I'll give my two cents on this very interesting and complicated topic. I do not claim to be an expert.
First off, "secular nationalism" is very difficult. It is so much easier to use religion as a basis for a national identity. Now the country might call itself secular but the bedrock is the religion, and the culture flows from that. So many ethnic groups in OTL used this kind of nationalism to I feel it is kind of how ethnic nationalism works. There are a couple exceptions, like the Germans, but so many countries use religious identity as the basis of their identity, it is kind of a rule.
It terms of identity, the Greeks have a historical continuity going back thousands of years, but I feel that "modern" Greeks have a restricted sense of identity, we might call this a "Athenian Identity" and contrast it with a "Byzantine Identity" or a "Macedonian Identity." The most familer Greeks to the western world were the Athenians so that was what western-educated Greeks focused on when they regained their independence. However, to a Greek in Byzantium, being "Greek" was so much more than just being Athenian. When the Greeks regained their independence from the Turks, the nebulous definition of "Greek" started to be used in contrast to the Ottomans. Being "Greek" meant being a Christian, so a lot of Muslim Greeks got alienated and went along with the Turks.
To create a secular Greece means avoiding alienating the Muslims, which requires, I am sad to say, forcing their hand by destroying the Ottomans. Entirely, suddenly, and all at once. And in the ashes creating a new Byzantine Empire where every Greek IS a Greek, regardless of religion. That is the hard part, because it it is difficult to get all the Great Powers of Europe accepting of that. The "Greek Plan" by Catherine the Great is a example that had some promise but it was becoming less popular and feasible after 1800 rolled around. However, it is not ASB that Alexander could have implemented his Grandmother's plan had the circumstances lined up in Europe.