I can understand how learning another language teaches a lot about a culture, I speak two languages other than English myself.
So...that leaves me with a couple of questions. Specifically, how can a person learn about the culture of a people using the limited vocabulary of MA words? And what specifically is being learned?
I think if a person is interested in studying the language as part of the art, and as a means of learning more about the culture of the people who developed that art, they are probably not limiting themselves to a MA vocabulary. I certainly am not. Am I fluent? Not in the least. Nor am I ever likely to be. But I'll learn more than I know now.
As to what, specifically, is learned, I don't know that I can answer, given my current poor level of knowledge. Sure, there are things I've learned that I consider interesting, but I'm not yet sure what they tell me about the culture. For example, I've learned that what we consider a reverse punch is a regular punch in Korea. Or the different connotations of the word "Master" here vs Korea.
When I travel, I make an effort to learn at least a smattering of the local language, rather that expecting the locals to speak English. At least enough to greet people, ask simple questions, and thank people properly. I consider this respectful and just plain common courtesy. I spend a lot more time on Tae Kwon Do than I do traveling. And I expect to (eventually) know much more of the language, culture and history of the people who developed this art than I do now.
Please note that I am in no way saying that this is required to learn a martial art. I'm just saying that I personally find it interesting and useful. And if my saying "hai san" at the end of class, rather than "class dismissed" sparks an interest in another student, then that cannot be considered a bad thing.