...Suddenly they are treating you as a normal Korean teacher or mentor and not as this weird white person that just plays games with us twice a week.
Hmmm, rosy dreams abound...I think it's great that you want to resonate with them, but IMO we shouldn't be treated as a normal Korean teacher. I have issue with your assumption that other NETs are treated as weird white people...especially because I personally am not white...or a person...I'm actually a brown hamster that enjoys jumping up and down on keys as I've disposed of my human master ages ago...anyhow on with the point.
I'm not sure of your angle in class, but we are here to improve their English. Yes, they struggle with expressing themselves, but the idea is to help them through that frustration and communicate effectively. I don't know if it's naivete or what, but obviously people will 9 times out of 10 revert back to their native tongue. You shouldn't enable it though.
I study and speak Korean actively as much as I can, but NOT in the classroom. I use it sparingly or to recapture their attention. If you get kids into a pattern or habit, you can just about use any hamster squeak to use discipline and they'll get it. You don't have to speak Korean, even if you really want to.
If there are deeper inherent issues of wanting to live and be treated like a Korean person, good luck to you. You'll need patience, understanding, and an incredibly long fuse. It's not impossible, but if you're here for a year or 2 for adventure, forget about it. However, this sounds bold, but it has no place or bearing with them learning English more effectively in the classroom.
None.Bottom line is: Don't strive to act like a Korean teacher, because surprise surprise, they have plenty of them who are better at acting like Korean teachers than you. It's harsh, but once Korea finds out that they don't need NETs here, they'll get rid of us. Wanting to be treated like a Korean teacher won't help. This is simple business-sense. They employ us because we offer something different; if we act the same, what's the point of having us?
Be careful with how you approach them in class, because your intentions of reaching common ground can turn into the "Look at how well I can speak Korean" show.