Have you thought about what is Korean Beauty?
70% percent of Korean peninsula is mountain. The majority of trees is pine tree.
One of the famous Korean Beauties is Pine tree on the rock. It had survived on the rock enduring the bad weather.
I see almost all people go to the Palace, Insadong, Cheonggyeocheon stream, N Seoul Tower, etc. I highly recommend you to visit one of mountains if you have more than 4 hours in Korea. Because there lots of mountain those will take from 1 hour to 12 hours for trekking.
Even there are many parks in the foot of mountain. All Ajumma(아줌마, married woman who generally have grown-up children and in their forties or over) are there. If you understand Ajumma's style, you are understanding half of Korean culture.
Just forget these things.
Take a deep breath in the any mountain.
These mountains are just near by all of your hotels.
It is very beautiful even though it is not famous. But it is real nature.
let's go back to the nature in the mountain. Jest at the edge of the mountain, it is enough.