It occurred to me, while reading the Philosophical Investigations, that a potentially helpful analogy for understanding Wittgenstein's notion of language as "instrument" is Thomas Aquinas's defense of speaking of the Word using Jesus's human nature "instrumentally." (I suppose the helpfulness could extend in the other direction as well.) For Wittgenstein, language isn't an instrument because it's an item in our world external to us, ready for use in executing goals unrelated to language, that is, as a tool that could otherwise be accomplished equally well without language. (Not: this hammer will do, although that screwdriver would work, too.) Rather, language is an instrument because it performs work: we should keep ourselves from asking about something other than language that is the "real" thing happening or being communicated above or behind or within the language. When talking about meaning and communication, we should simply ask what the language itself is doing—what the words are up to in shared discourse. Language as a feature of human life is pragmatic; it fulfills tasks. If you want to know what the words mean, look to see what particular people are doing with them; observe what they're using them for.
Which isn't to say, as one might take it to mean, that language is not constitutive of human life and being. It unquestionably is. But its centrality to human sociality and existence is not to the exclusion of the practical, even mundane role it plays. Language, in other words, isn't a hammer next to the screwdriver. It's the toolset we're born with and into, which we receive and are trained in the exercise of, and which we develop and refine over a lifetime.
It seems to me that Thomas's understanding of the relationship between the divine Word and Jesus's human nature is quite similar to this account. Thomas argues that the divine incarnate Son's human nature is indeed constitutive of Jesus—who is a divine person enfleshed as a human being—precisely because he would not be human apart from it. But the human nature is nevertheless "instrumental" to the Word insofar as it is a medium of the Word's action and presence in the corporeal world. In this way it is neither accidental (because he wouldn't be human) nor essential (because he either wouldn't be or would cease to be divine), but rather—following the Word's assumption of human nature, resulting in a union in the hypostasis of the Word (not in the divine nature)—subsists as constitutive of the man Jesus of Nazareth, who just is the eternal Son in the flesh. Just so, because the human nature, which includes the soul and body of the man, is the Word's own, it is the instrument of the Word's agency in human, physical, material life. When Jesus acts, it is the Word acting, that is, acting in and through the instrument of its own (assumed) human nature.
In short, language is to human life (for Wittgenstein) as human nature is to the incarnate Word's agency (for Thomas): a not-quite-essential, but nonetheless-not-accidental, instrument.
(Actual scholars and experts in these figures, commence your corrections.)