As I understand you reasoning, man did not have pottery till 5000 years ago.
Perhaps that is the source of your error.
Well, traditionally, the earliest evidence for pottery is from Ubaid Mesopotamia, about 5000 years ago.
Yes, there have been more recent claims to date other finds earlier than that by carbon dating, but I don’t believe the totality of the evidence actually supports an earlier dating.
For example, carbon dating of pottery fragments from Japan may say “15000” years old but we know the Japanese culture was backwards compared to the Chinese until very recently, so it’s simply preposterous to think that:
a) this backwards culture independently invented pottery before their more advanced, literate neighbors
and
b) this backwards culture invented pottery and somehow it wasn’t transmitted to China despite constant contact between the two cultures.
That doesn’t make any sense. Therefore, the more likely explanation is that the carbon dating is wrong.