This is mostly not a position discussed in linguistics (it is sort of identifiable as an application of nihilism in philosophy), but on occasion – this is one – linguists can offer reasons to reject the claim.
The main linguistic attack would be against the premise that there is a problem, the claim that "all languages are inadequate for every purpose languages are assigned (thinking, communicating, etc.)". The two fatally-weak concepts therein are "assigned" and "inadequate". In lieu of a theistic premise, languages aren't "assigned" (by who?). Language is a tool, just as a hammer is a tool. It is up to a person in possession of the tool to use the tool for a given purpose. Typically, hammers are used to drive nails into soft substances, but they can be used to break up hard substances, or used to acquire food ("I'll trade you this pizza for your hammer"). You might imagine some idiot nihilist rephrasing the claim as "for any imaginable purpose", but I successfully insulted the nihilist using language.
The nihilist, of course, has to posit that his idea cannot be conveyed to others, since you certainly can't convey the proposition non-linguistically, so in fact if the nihilist is correct, he never communicated the claim to anyone else, nor did he think of the claim (using what? Language!).
The nihilist may then retreat to the position that his attempt to express the idea has some form, but it isn't adequate. Again, common sense tells you that people can use language to achieve various ends. We know that Bene Gesserit Voice is total fiction, so you cannot overcome free will linguistically, but this is not a fact about language, this is a fact about free will.
The most attackable fact about language that suggests "inadequacy" is "ambiguity", the fact that a given linguistic object can refer to more than one imaginable state of affairs. For instance, the verb "die" might be assigned a particular definition such that living things ceasing to be alive is "dying". And yet we do say that "my phone just died", worse yet, that could refer to permanent non-functioning (bricking) or temporary non-functioning (battery ran out).
The problem here is that language is inadequate from one perspective (micro-precise description of a state of affairs) and totally adequate from a competing perspective (identifying a functional similarity between two states of affairs). Indeed, a simple word like "dog" is both perfectly adequate (in unifying disparate individuals) and totally inadequate (in allowing the unification of clearly different individuals). Adequacy is meaningless without a purpose: the nihilist failed to consider competing purposes.
The basic flaw in the argument is the attempt to lay blame on language and to ignore the nature of reasoning. I hereby posit a radical version of subjective idealism: "Only The Mind exists". Feel free to try to disprove this: my retort will be "Only The Mind exists". Perhaps I might say "Nothing exists besides The Mind". I have now analogously proven that "reasoning is inadequate for everything".
As you might guess, linguists generally don't take such nonsense seriously. The linguist's response is "Your problem is not with language".