Those signs have nothing to do with protecting culture. Most of the time, they are simply a naive defense against dealing with a population often doesn't speak Japanese very well.

the US has that much history too, it just chooses to ignore it

Japan also closed itself to the world for 250 years, and it looks like some people haven't still gotten the memo that the Edo era is long gone.

>USA has a 250 year history while Japan has 10x of that.

Is that supposed to imply that Japan has more culture, or that it needs more protection because it's 10x longer? Even if Japanese culture is 10x longer than American culture, it doesn't necessarily follow that there's less of it. Pop music and hollywood music might not be considered "culture" by snobs, but they're still culture, and arguably more plentiful and pervasive than Japanese culture.