I think for a character that isn't even in the credits, Mrs. Kim is a very well developed character. They started with two stereotypes as a starting point. Korean and Christian. I'm guessing they didn't really decide on the denomination of Seventh Adventist until later. But, gradually, Mrs. Kim grew into more. She's a business woman, and a mother who is strict out of protectiveness of her daughter. I loved the episode where she grounds Lane, and after a talk with Lorelai, she lets Lane out just to the front of the yard. Comic way of showing that Mrs. Kim isn't totally bound by the initial stereotype, but can soften. And, in later seasons, we've seen her soften more, but (for the most part) still stay in character.
If Mrs. Kim or any character that they spent a lot of time on was nothing more than the stereotypes, that would be a problem I think. Maybe why I don't care for either Taylor or Kirk. I'm not exactly sure what Kirk is a stereotype of, (maybe Kirk should be a stereotype in itself) but he's annoying. But, for the most part, the characters all go beyond just one stereotype, into being 3 dimensional. And, if parts of those dimensions include stereotypes, that's fine by me. I agree that everyone has stereotypical traits about them. Stereotypes aren't bad by definition. It's when that's all that's shown, then it's a problem.