Those buses are still from the "quarantine days." Uber exists, but they're just regular taxis registered to the service (just like Kakao T and many other similar hailing apps). They're not private drivers. The airport is also literally on a detached island by itself, so you won't have many taxis roaming around waiting for pickups like say the US.

It's VERY EASY to get a taxi - walk outside to the taxi line and there is a section for Seoul taxis.

There are actually SOME buses running to Gangnam as you found on the website, but it might not get you EXACTLY to your hotel. Just take a taxi.

I live in Gangnam and can tell you 100% the meter rate, including the highway toll, is around 60,000W to Gangnam Station area no traffic. So, to COEX, expect 67 or so, add some time for traffic, you're looking at like 70-75,000W.

Do NOT let yourself get hosed a few extra bucks - if your driver tries ANYTHING, report him immediately to 1330 tourist helpline. If he gets reported more than once, he loses his license. Big crackdown now by govt that tourism is coming back. Not all taxi drivers are a-holes, but just like every city, it can happen.