However, if we hypothesize that in similarity how the ancient macedonians started becoming roman and byzantine after, the same way they were actually assimilated into being slavs and mixed with them, because the slavs were in fact very open minded who was a slav and who wasnt, then we could say that the genetic line is there and in fact lots of people in the balkans and anatolia and south italy trace greek heritage.
The slavs from russia and ural are very different looking than the balkan slavs. In addition, you dont see many ethnic hungarians that look asian, even though the initial magyars were asian nomads. Same with bulgaria, the bulgars were asian nomads from the steppe, but hardly any ethnic bulgarians look asian. Meaning, the initial conquerors were a minority group being overlords or leaders of a majority local natives, who then got assimilated.
The mixing of peoples was there. The genetic trace is there. But the political rhetoric is crazy.