The point, which you seem to be missing, is drawn somewhere on the arbitrary line between "true" Japanese culture, and foreign culture. Buddhism isn't Chinese, but it reached Japan by way of China and Korea.
So, yes, it's a Japanese Buddhist/Shinto temple. But the religion isn't originally Japanese, and neither is the architecture. The only thing native there is the Shinto.
No one considers modern Japan to not be Japanese, despite it being, in some sense, an American reboot of a Prussian remix of Imperial China. If you want to say Japanese culture predates the US by "thousands of years," fine - except you would need to ignore the thousands of years of European culture that America is based on to do so.
And also ignore the fact that, yes, modern Japanese culture is very much a product of the postwar period, making it both older than, and newer than, the US. And the aspects which are newer are likely the more relevant to Japanese people.