Turkey makes a mistake of trying to deal with it.

Turkey's public education in this regard, much like most of the policies about Turkey's founding and identity, is primarily defensive in nature. It tried to justify Turkish presence in a combination of glorifying Turkish conquest, minimizing Greek history in Anatolia and emphasizing or even making fanciful claims about Pre-Hellenic presence. Because it was primarily trying to discredit the Western-backed Greek claims over parts of Anatolia.

This might seem strange now what with different ideas about land grabs but back then it was a real concern and anxiety of non-Western states to try to justify their existence on a piece of land and make arguments for why their presence was not just defensible but actually desirable. Because being delegitimized on the ground of historical claims and being colonized as a result was not only not uncommon but the standard as it happened in most Western colonies.

Classical antiquity was very popular in the West and any presence that could be justified in the texts of classical antiquity was prized. This is also a reason for all sorts of bizarre theories of origins and claims in Balkans about ancient past that claims descendancy from one of the groups identifiable in classical antiquity. This is also why a lot of association was made in particular about Hittites. After all peak of Hittite state predated Alexander the Great and subsequent Hellenization of Anatolia for about a thousand years, that's roughly the same "few centuries" as Turkish presence in Anatolia.

It's unfortunate that his was something to not only "deal with" as a society, but also something to prove in international relations. Because historically Turks didn't really reckon with this in any way, some of the more educated acknowledged that many who call themselves Turks or similar were of mixed origins and often had non-Muslim ancestors only a generation or two before them. It doesn't seem to have caused them any problems. I assume same was true for any Hellenized Christian that lived in 6th century Anatolia whose ancestors few generations ago might have not spoken Greek or be a Christian.

Greeks of course, despite making the ground zero of "Western civilization" in the classical text weren't spared this. Because Eastern Roman Empire wasn't exactly favorable in West, it was something that Westerners measured their success against and they never particularly liked any competition over their own claim over Roman Empire. That's why modern Greeks, despite 1000 years of Eastern Roman history from collapse of Western part to end of Roman Empire in 1453 primarily attached itself to Greek antiquity. Those few centuries making the 1000 years of history weren't seen as important as the long Hellenistic period from Golden Age of Athens to Roman conquest, which was 500 years long, certainly longer than few centuries that fits in 1000 years.

The reason for this is of course because people aren't simply just a sum of genetic or cultural sediment that forms into a singular whole that can identified without any connection to anyone else. Yet nation must be singular, independent and never conquered even if its territory is taken over. Admitting connections or worse influence by another people who speaks another language is admitting defeat, as bad as being conquered.

That's why, when Turkey tried to deal with it, had to discredit any connection the Central Asian conquerors could have had to Anatolia and its history, even when the religion got converted to Islam by Iranians whose history itself was deeply tied with Anatolia and Alexander the Great. Nor could it acknowledge that mentions of Greek, Persian, Arab and Indian figures, as well as Alexander, exists in both secular and religious texts of Ottoman Empire. Same Alexander that is sometimes identified as Dhu Al-Qarnayn, or Zülkarneyn, in Quran.