I personally know a Korean Methodist lay-leader, a woman, who serves on a Korean Methodist national board. I should sound her out on your many comments regarding South Korean Christians. They are certainly not the most conservative Korean Christians, though most are more conservative, even in their Christianity than U.S. Methodists. They held onto the faith values they learned from earlier generations of Methodist missionaries, while the U.S. descendents of those missionary generations abandoned many of them.
What follows is a gross oversimplification but is correct in general outline.
Unlike the situation in the United States, the major denominations in South Korea are Presbyterians, Full Gospel (which is equivalent to the Assemblies of God), Baptists, Methodists and Roman Catholics, in roughly that order. The Presbyterians are divided into a number of different Presbyterian denominations, the biggest of which are two large evangelical denominations of about 2.2 million members each, one much more conservative denomination about a fifth of that size, one somewhat more conservative denomination about a quarter of that size, and one quite liberal denomination about a tenth of that size.
Other denominations exist in Korea, many of which trace their origins to the post-World War II era, but those are the largest and most influential groups.
The Roman Catholics trace their heritage to the work of Koreans who spent time in China and were the first to introduce Christianity to South Korea — I'd have to do the research but I'm fairly sure it was in the late 1600s or early 1700s, and certainly long before anyone else. They obtained the help of French missionaries, some of whom were martyred in Korea, but were not particularly successful until the large growth of Christianity under the influence of Presbyterian and Methodist missionaries who began their work in the late 1800s. The Baptists and the groups that became the Full Gospel churches came somewhat later.
Protestantism entered Korea while the country was still independent. However, the Japanese obtained a protectorate in 1905 and took over the country directly a half-decade later, ruling it until 1945, and by the 1920s were severely persecuting Christians who showed disloyalty to the Empire, especially with regard to refusal to participate in Shinto shrine worship of the Japanese emperor.
The four main splits in Presbyterianism date back to the expulsion of the Japanese following World War II, when the most conservative Presbyterians refused to have anything to do with those who had participated in emperor worship. The main Presbyterian body expelled a group of liberal denominational leaders who had collaborated with the Japanese and had adopted German higher-critical liberal views; that main body then divided over whether to cooperate with American Presbyterian missionaries in the PCUSA (Northern) and PCUS (Southern) denominations, or whether to work only with the Christian Reformed Church, Orthodox Presbyterian Church, and other strictly conservative Reformed denominations, in large measure because under Japanese rule an Orthodox Presbyterian missionary had nearly been killed by the Japanese for refusing to participate in emperor worship at the same time that most of the American missionaries were willing to do so. The initial seceder group merged with the more conservative of the two larger bodies, but the merger was not successful and part of the initial seceder group split again to form a smaller denomination which now has ties with the Orthodox Presbyterians, Canadian Reformed, and GKN-vrijgemaakt in the Netherlands.
I do not know enough of the history of the Methodists in Korea to speak with any level of authority, but my understanding is that the Korean Methodists, like the Presbyterians, suffered from problems caused by American missionaries who were too willing to compromise with liberalism. Unlike the Korean Presbyterians, the Korean Methodists were not able to find conservative Methodist bodies overseas after the Korean War who were willing to help them and that has not been good for the Methodists, who have been influenced more by American liberalism than the other major Korean denominations.
Again, this is a gross oversimplification, but I hope it is of some help.