> Also why do people trust the government when it only became a democracy about 30 years ago, and things like this happened?

Considering that the country's entire history span about 70 years (excluding the provisional government; the provisional government is exactly 100 yrs old in 2019), 30 years is a long time.

Presidents are elected with direct election; a direct consequence from the Park Jong-chol incident you have mentioned, and 1987 is the year of South Korea's 'real' democracy.

People have the power to impeach presidents with peaceful protests[0][1] and people are proud of it.

South Korea's democracy is highly trusted in the country, IMHO.

[0] https://en.wikipedia.org/wiki/Impeachment_of_Park_Geun-hye [1] https://www.bbc.com/news/world-asia-39227342