I would say you are getting into identifying a big issue in applied epistemology more generally namely that cross domain communication and knowledge transfer can be very hard which is to say that asking those trained in economics about the applicability of a mathematical proof to their domain is likely to yield unsatisfactory answers to the same degree that say, asking neurologists about the impact of some aspect of quantum physics is applicable to their understanding of neurological processes.
Sometimes even more worryingly the concepts may have been independently considered in two fields and while they conceptually discuss the same issues they may use entirely different vocabularies to describe this same concept. This occurs even amongst comparatively closely related fields of study like electrical and mechanical engineering. A particular example is that of control theory that spans these two fields and for various historical reasons have taken curious diverging paths at times.
That is not to say that the connections don't exist but the answers that make these questions interesting are not easy to come by as the concepts in each field can take significant study within the respective fields to even master and truly appreciate thus limiting the time available for experts from either field to really be able to discover the ideas from fields outside of their own even if there is considerable overlap never mind to meaningfully translate between the vocabularies to engage with those outside their field.
Is there a connection between the broad class of related ideas represented by the likes of Gödel's incompleteness theorems, Tarski's undefinability theorem, Church's answer to Hilbert's Entscheidungsproblem and finally and maybe more famously the Halting Problem, and more broadly all of human epistemology? I can't claim to be an expert in any of these specific proofs but I have to say there seems at least to me an intuitive sense in which this connection is unavoidable. They seem to point to some significant limits to what can and cannot be known and what it even means to know anything at all. Is the connection sometimes maybe overhyped and sensationalized a little by certain groups? Again I think the answer is yes but I think the resultant hesitancy in fields outside of those where these proofs first originated to investigate their consequences on their own fields is also a mistake as these are extremely significant and rigorous findings.
If highly simplified and formal systems are plagued with inherent contradictions, and computational limits then it does not bode well for larger more complex systems systems of knowledge. Sometimes practitioners unfamiliar with these proofs defend against this issue when first exposed to these ideas by making claims that these theoretical shortcomings do not translate to any limitations in practice but I would say that at the least computer science strongly refutes this proposition as the implications of the halting problem and other computability problems are quickly and trivially practical not least to massive economic consequence.
As for how these ideas have been addressed within economics I think there does feel like there are some similarities between the class of proofs previously listed and the Mises-Hayek economic calculation problem though I'm not familiar with any rigorous attempts to connect these ideas despite their passing similarities which as I initially mentioned seems related to the the difficulties encountered in sharing knowledge across domains.
I would therefore say that any attempt to reject the impact of this class of proofs on other fields is horribly misguided and smacks of significant doses of ignorance, but this must be weighed against the reality that in many practical fields the ability to make positive progress is not impeded by their applicability.
Not knowing and stated more strongly maybe never actually being able to know does not mean that significant value and progress cannot be achieved. After all, we aren't comparing the correctness and consistency of our knowledge systems to some system created by some perfect oracle somewhere but merely to those created by other humans similarly limited by the implications of these theories.