Now this is part of the story but perhaps not the whole story. That part of language most intimately bound to culture is its vocabulary, its inventory of words. All societies develop terminologies for naming kinfolk, social and political institutions, aspects of technology and the economy, features of the environment, and so on. Terminologies develop to greater or lesser degrees depending upon the society's interests and its ways of fitting into and exploiting its environment. Vocabularies reflect the importance of these activities to the Iroquois.
Comparable situations exist for all cultures and languages.
It is here that we would say that languages have a certain independence, a certain life of their own. And indeed if you look into the history of linguistics, you will find that there is a somewhat darker side to this question. At one time, before the period of what we would call scientific linguistics, people liked to classify languages according to a few basic structural types, and these types were thought to correlate with stages of cultural development. Such ideas are inherently racist and seem silly to us today, but I mention them as examples of the kind of pitfall one can stumble into by taking a too literal approach to the relationship between language and culture. Closer to home, the Northern Iroquoian languages are, as you know, related to Cherokee, in the sense that at some distant time in the past, perhaps around 4,000 years ago, there was a single language that was their common ancestor.