I'm pretty sure it would have had some presence in Western Europe (in terms of curds and whey).
For E Asia you need to have the idea of using dairy products to take root in the first place. I'm not sure why it didn't- East Asians do have more prevalent levels of that lactose intolerance gene but so does everyone else who isn't Western European and it hasn't stopped Arabs, Persians, Indians etc having dairy based diets.
It's especially strange considering how popular dairy products are today in East Asia - Young people in China drink milk and yogurt religiously, and lately there's even a "make your own yogurt" fad going around that popped up almost overnight. I believe Hokkaido is also now famous for its dairy - At least, I've seen a lot of Japanese prepackaged breads made with "real Hokkaido milk".
Yet, dairy in China is still seen as either a minority food or a foreign import. The most famous dairy company is Mengniu (Mongolian Cow), and the most traditional Chinese dairy products all have origins in border regions - Milk and yogurt products of Inner Mongolia, rubing and rulao of the Bai people in Yunnan (which seems to be a Chinese take on Indian paneer), Cantonese milk pudding (distinctively Cantonese, but clearly adapted from European influences), and Beijing palace cheese (seemingly from the Manchu Qing, but I could be wrong). Here in Xinjiang, there's katiq, kaymak, ayran, suzme, qurut, and many other dairy products that have many names in the minority languages but lack such distinctions in Chinese.
You'd think at some point dairy would have spread from the Silk Road to the rest of China - The Mongols didn't introduce it during the Yuan dynasty? I go to any supermarket in China, and the local commercial brands of yogurt all use artwork conjuring up the grasslands and minorities of Inner Mongolia and Xinjiang... There's also the new, overpriced, fad yogurts that claim to be foreign specialties - There's a new Chinese brand of Greek yogurt displaying Athenian architecture, an older Bulgarian yogurt displaying a woman in traditional clothes, and a slightly newer kefir (开菲尔) displaying another nondescript ethnic woman, perhaps Azeri. They're always in the same packaging, they're always twice as expensive as other yogurts, and they don't taste anything like the equivalent products found in Western countries - In fact, all three taste nearly identical after I did a taste comparison test.