Wilson Tang, who owns the storied NYC dim sum house Nom Wah, considers Chinese treatment of cows to be a main, understated force in the lack of dairy in China specifically.
"Cows were traditionally used as tools for work," he said, "and oftentimes villages would have very few animal resources at hand. So they couldn't exactly use animals they need for farming purposes to create milk for cheese."
Which makes sense. But the biggest reason Asian cultures don't regularly incorporate cheese into their cooking is probably because so many East Asians are lactose intolerant. In fact, they're drastically more likely to be lactose intolerant than Westerners. And so many East Asians are lactose intolerant because of a lack of exposure to cheese. It's a vicious cycle. But it started somewhere.