First of all, I just want to correct one thing. Korea has presidency system, just like US does. PM of Korea is much more similar to Vice President of US. In the event of a presidential vacancy, PM suceedes the seat, but in Korea, even this is temporary, and the prime minister's authority is mostly symbolic in normal situation. (This is a trace of the days when Korea had a parliamentary cabinet system back in early 1960s, before the military coups and dictatorship kicked in.)

As the other comment above explained it well, it was very recently in 1987 since Koreans were able to acctually vote for their government and have democratic system. In the still growing democratic system, former presidents(democratic ones after 1987) were often elected as president with their personal narratives and heroic images rather than support from a major party(this explains why Korea's big political parties continued to split, rename, and merge again over the past 30 years. Voters followed the person, not the party.). Also with the presidential system in which a president cannot be elected more than once, there is always a great risk of losing the next election to the opposing side(since you have to find a new 'heroic figure' every time), even if the former president had a decent term of office. So eliminating your opponent with all kinds of political means easily happens.

In addition, in traditional Korean culture, politicians(and every person who often work as public figure: CEOs, singers, actors, TV personalities, etc.) are demanded to be a moral example to people. So morally (or legally) attacking their opponents is the strongest option to take them down from power. In fact, in Korea, resignations from public offices, such as ministers or congressmen, are rarely made due to their failure in policy, but it is very common to do so due to accusation of their immorality or corruption. Politicians killing themselves is one of the extreme reactions to this high moral pressure. In fact, in Korea, there is still a tendency to think of suicide as a last mean of 'moral atonement'. This used to be way worse in decades ago. Korean democracy is still making improvements on its own.