There hasn't even been 5000 years of western culture...
https://en.wikipedia.org/wiki/History_of_Indigenous_Australi...
Yes, the glorious Western civilization did the same pretty much everywhere, but Australian Aboriginal people's culture is unique due to their very long period of isolation - it's a gem and wonder worth studying and learning from. Yet, the Western instinct was to trample on and destroy it, systematically, irreversibly, just for the sake of it. Reading the Wiki page you linked some time back literally left me in tears. How could we. I don't even.
Co-opted first by Alexander, then Caesar. Later Napoleon also dropped by ...
Seems to have slipped everybody's mind here, strangely.
This is like claiming the Islamic Republic of Iran is the same culture as ancient Persia.
But if you go back to the Early Greek era of the Minoan civilizations then it is about 5000 years.
It's all slow, gradual accretion though, so you can draw the line in lots of places.
Either way, on any of those benchmarks Mesopotamia precedes it by at least a few centuries or more, though Minoan does get you back pretty far. I don't know the migration patterns, but I think there was influence from Mesopotamia.
Classical Greece ~2500 years is probably often where the line is drawn because that's where some of the foundational works of culture, philosophy, and math had their start and became founding members of the "Western Canon". For millenia, students were trained in Greek and Latin and read the great works of those time periods. But there was a civilization that gradually evolved into that time period, complete with it's own Dark Ages before things climbed back up to what became Classical.