I asked this question yesterday, and edited it today to include clarification.
It is - I think - a fairly honest and straightforward question, and I have provided a detailed explanation of my thinking within the body of the question.
Perhaps I have made mistakes, but as far as I can tell the key points of my explanation are consistent with available literature on differential geometry with the exception of disallowing exceptions to the rule of either distinguishing between or identifying isomorphic structures (the latter of which I believe is more conventional, and so have chosen to employ.) I have highlighted the relevant points:
The consistent application of the convention of identifying isomorphic structures
Why the consistent application of convention matters
The chain of implications leading to the independence of the tangent space from smooth structure
The distinction between homeomorphism and diffeomorphism
The inadmissibility of the tangent spaces distinguished by points while retaining the "intrinsic" nature of the tangent space
The possibility of defining the tangent bundle to facilitate the transfer of smooth structure, and the possibility of defining it as a topological invariant
I am quite certain that I am incorrect in my thinking as well, but that is what the question is about, and none of the responses - in comments or the one posted answer - have actually addressed this.
Is there an issue with the content of the question beyond my simply being incorrect about some aspect of differential geometry? Is there an inconsistency in my reasoning? Is it my tone? How can I improve the question?