What we do have is actual written characters dating back to ~1200BC. The earliest Greek texts that we have date back to ~1400BC. So based on this evidence you could say Greek was earlier, but not by nearly as far as you suggest.
No, you just don't understand.
The Greek script has been in continuous use in its modern form since about 1400BC; the modern Chinese script dates back to maybe 200-700AD. The earliest writing systems in the Middle East go back to at least 3400BC, 2200 years before any documented Chinese writing (oracle bone writing, a rather restricted form). In fact, it is far more plausible that China got writing via the Silk Road than that the Chinese invented it independently.
Modern Chinese society is based on the Qin culture which dates back to at least 9th century BC. ... While this is not terribly old, the unification under the Qin emperor in 221BC wiped out all of the other cultures of the Chinese people which dated far further back.
And then it split again a few centuries later, then there were civil wars, rebellions, more splits and unifications, religious divisions, even more splits, etc. Not much different from what Europe experienced with Rome and afterwards. But parts of Europe have a history that goes back much further still.
As far as contemporary cultures are concerned, there are very few that can date back as far as 200BC.
Well, at the scale and fuzziness at which you define a "culture", we have four major surviving cultures: China, Europe, India, and the Middle East. Of those, China is the youngest.