Your bags don’t go through the scanner or anything at customs, so you’re right that no one will probably notice. However, I recommend declaring because they can confiscate it otherwise. It literally only takes 5 minutes to declare it. (maybe more if there’s a line).
I brought $16,000USD with me in $100 bills so I had 2 envelops filled with cash in my fanny pack. On your flight to Korea, you will receive a customs form. Check “Yes” to the question if you have over $10,000 in cash. This is literally all I did. I had no other paper work aside from the customs form given to me on the plane (or can be obtained near customs if you fall asleep).
When I arrived at the customs (Gimpo airport), I gave them the slip. The officer told me to go to the left where it had more security people. I approached the more specified customs person, and she just asked me the reason and asked me to show her the cash. She didnt count it or anything. Literally just a 5 second peek, and she laughed whole heartedly when I told her I was going for cosmetic surgery. She marked some paper. I then returned to the first customs officer and gave them the paper. Super easy and they’re nice about it. I can’t imagine having trouble unless you’re bringing an obscene amount of cash.