You’re completely correct in the sense that the West or Western, in the sense of a cultural identity as opposed to a geopolitical alliance (although materially in the modern world these are ofc connected), is not only modern but pretty artificial (though real). However the appropriation, which ran through Roman culture to modern Western culture, still indicates by itself that modern Western culture was and is very influenced by Graeco-Roman civilization. Although a lot of this appropriation is also explicit, retroactive and indirect, and based on idealistic attempts to identify themselves with supposed aspects of ancient Greek culture which don’t actually hold up on scrutiny.

As you’ve said, Ancient Greece had far more in common with the other Mediterranean cultures of the time than the modern WASPs of the West or the ancient Germanic or Celtic peoples. Like if an ancient Greek teleported to the current era your average yankie WASP would deffo not think they’re hwhite like them. This is also ignoring the fact that whiteness as a category as it emerged in early modern Europe and America did not exist for the Greeks, who would also have seen themselves as having far more in common with, say, the Egyptians or Phoenicians than with the ancient Teutons.

Then again WASPs already think the same thing about many modern Greeks, joking that they are basically Turks. There is a bunch of cultural things that are shared due to the Ottoman period, but we all known what they mean when they say this (i.e. the joke that southern Europeans are only semi-hwhite).