Why do most people focus on ancient Greek history ignoring the rest of the Greek history?

The West claims its patrimony from the Renaissance West and Mediaeval West. The Mediaeval West claimed its patrimony from Rome. Rome, and the Renaissance West, claimed their cultural patrimony from Ancient Greece.

So Ancient Greece matters to the West, because the West regarded itself as the cultural inheritor of Ancient Greece.

The Byzantine Empire was not regarded as the cultural ancestor of the West. It is regarded as the cultural ancestor of Russia, which is why you can find a lot of Byzantine Studies PDFs on dark corners of the Russian internet. But the West has no more reason to pay attention to Byzantium than to the Abbasids or the Safavids: an important empire, but not their empire.

Ditto Modern Greek history. If the West doesn’t pay that much attention to Ottoman history (not their empire), it will pay even less attention to the Greek elite within the Ottoman Empire. And Modern Greece is just another poor southern European country.

Now, I’m Greek, and Mediaeval and Modern Greek history is incredibly important to me. Mediaeval and Modern Greek history is also important to many Westerners (Byzantine studies is not a Greek-only or even Eastern European-only affair), and I’m grateful for it. But there are sound cultural reasons why Byzantium is of more interest to the average Russian than the average French.