I think the question might be better worded as "why has Chinese medicine become so well-known internationally, and not Korean/Japanese/Vietnamese?"

As others have mentioned, traditional medicines continue to exist almost everywhere. Indeed, in anthropology, the subfield of medical anthropology is increasingly in demand, and within medical anthropology, there's a whole cottage industry of studying the interactions between local, traditional medical ideas and modern, Western medical ideas. In Turkey, where I do research, the best known is We Have No Microbes Here: Healing Practices in a Turkish Black Sea Village by Sylvia Wing Önder, but there's also another one called New Organs Within Us: Transplants and the Moral Economy about organ transplants specifically.

Offhand, I can think of any about Korea or Japan (or China, for that matter), but the most commonly assigned book from this genre is The Spirit Catches You and You Fall Down: A Hmong Child, Her American Doctors, and the Collision of Two Cultures by Anne Fadiman (interestingly, a journalist, not an anthropologist, so it's better written but lighter on theory). It's about a Laotian Hmong refugee family in California who have an epileptic daughter. You might find the book very interesting. Here's its Wikipedia page which gives an extensive summary. If you're curious, I'm sure you can find medical anthropology books about Korea or Japan which give more detailed findings from those areas.

Edit: for example, here's an article about the standardization of traditional medicine in Korea, and it's integration into "modern" (Western) hospital setting there's. They call it "East Asian medicine" to emphasize the commonalities between traditional medical practices in the region. "East Asian Medicine in South Korea" (2012). Here's a summary of another article about "fire-illness", a depression-like psychiatric malady. Here's an article about traditional Korean medicine being done in Kazakstan.