Differences there in what's considered right--or maybe more, what's considered important to figuring out what's right. Like for me, the big thing would be that I learned the secret by accident, he's clearly keeping it secret, it's none of my business, so I have no business telling anyone. I would consider it his secret, and so the important thing would be that it's his and he didn't tell me, therefore it's not mine to tell. Even if I thought he was doing something stupid--and frankly I wouldn't think that, I'd be on his side given how I would expect that family to react when they found out.
But for a Korean, what's important is that it's family--any secret almost belongs to the family, even if they don't know it yet. Chae Rok still doesn't consider it his secret, but in a different way--it's like he needs to let it go, like what's wrong is that he, an outsider, knows it even though the people who ought to be privy to secrets, the family, don't.
And it's funny, like I can see it operating and understand that it's a big pressure and even a difference in the internal feeling of what's right and what's wrong, but I absolutely can't feel it in my gut--like, so what if they're family, I cannot make myself give much of a damn; if the old guy doesn't want to tell them, far as I'm concerned that's his business.