Anatolian languages died as a result of Hellenization policies that were forced upon Native speakers of Anatolian languages by Alexander’s generals, hellenic and hellenized elites and eventually the Church which declared greek to be liturgical language of the local Christians. Anatolian languages were supressed in favor of Greek. And Greek eventually absorbed all the Anatolian languages according to Wikipedia by the 7th century AD.
Contrary to the popular belief, they didn’t die out as a result of a Greek migration. There is not a single historic or genetic source which proves a huge Greek migration into any part of Anatolia. It is one of the most commonly believed myths that is presumed to be true. But there is not a solid evidence for it. Anatolia was never a part of Hellenic world. It was a part of Hellenized world. For the last 41 centuries, Anatolia has been inhabited by Anatolians and their partial descendants.
Anatolian Greeks are therefore largely same as their actual Hittite ancestors, they have little to no actual Greek heritage. If we’re generous, they are at most 5% Hellenic. Turks too retain their Anatolian heritage, albeit mixed with Turkics significantly. But neither Anatolian populations have any significant greek heritage. The other Native Anatolian population is Laz and they don’t seem to have any “Argonaut” ancestry either.