I've never been to Korea (either one), so my experience comes from Korean restaurants--mostly in China, if that helps. For example, last night's menu included many dishes in which cheese featured prominently: as a dip, as a dressing, etc. This is not an isolated observation; I've seen similar things at lots of other Korean places. Based on that (admittedly limited) experience, Korean food seems like an outlier.
Has Korean cuisine really embraced cheese more than other East Asian cuisines? Is this a recent phenomenon? Is this only true about Korean food outside Korea? (I doubt it's simply a matter of catering to customer demand, as Chinese food is virtually cheeseless.)
I'd say that your premise is flawed. Korean cuisine has not "embraced" cheese. Sure, there are lots of "fusion" places that have cheese ramen, or cheesy pork cutlet, but by and large, Korean dishes do not have cheese. Further, if we consider the fusion dishes, we're going to be getting within the 20 year rule, so there really isn't much to say about that here.
Now, if you want to talk about diet, the South Korean diet has definitely grown to include more dairy and meat products over the years, but it's going to be more milk and yogurt than cheese. This is simply a by-product of economic growth.
The antithesis of your observation would be my observation of Chinese food in Korea. We have Jjajangmyeon, a black bean sauce with noodles, or Tangsuyuk, fried pork with a sweet sauce. All of my students from China swear that this is not real Chinese food, and from my limited time in Beijing and Taipei, I know they're right.
If you'd like some recommendations for Korean cookbooks, you could take a look at The Korean Cookbook by Judy Hyun (1974). It's a little dated but as I said, if you're looking specifically for cheesy Korean dishes, they're going to be within the last 20 years.
Mongolia is located in East Asia and has always eaten huge amounts of cheese.