...and also Chinese Wall and Kung-fu...

CCP did everything it could to wipe out old traditions, and replace them with communism and cult of leaders. Latter they realized it is not enough to keep coherent national identity, and started promoting a few symbols they liked from past. The "big red door decorations" is not Chinese culture, it is a small fragment of it.

> It is simply absurd to claim Chinese culture only survived in "a few small places" in the mainland.

I do not claim that. It did not survived at all on mainland. "A few small places" are outside of mainland: Taiwan, Honk Kong, Macau and perhaps Singapore.