You're correct that kids at Eaton get 1 - 2 Chinese classes a week. They focus on language and culture. My daughter has been at Eaton for two years, and has learned enough to sing a few songs, exchange pleasantries and really impress waitstaff at Chinese restaurants, and catch a few words if she hears someone else's conversation. I don't think the limited words she has learned are particularly useful, but she is excited about learning a language and excited about learning about the world, which I think are totally the purpose. It's a positive, engaging environment and I think the chinese, however limited, adds to the kids' overall experience.
If you want your kid to have more exposure to the language, there are also after school classes - once or twice a week, I believe - for small groups. They are inexpensive and those kids seem to learn quite a bit more.
Overall, Eaton is a treasure of a small public school, and I hope that you enjoy.