NP-Complete Convergent Reductions?
Great question! I have also had thoughts along similar lines, but not found anything in the literature that fully answers all of these questions. However, for this part of your question, I think a satisfactory answer is given by Berman &amp; Hartmanis:


  what prevents us from adding some (polynomial amount of) structure to our NP-Complete problems so that reductions from one NP-Complete problem to another converge in size?


In some sense, nothing: Berman &amp; Hartmanis showed that if two languages have one-to-one, polynomial-time invertible, length-increasing reductions between them in both directions, then the languages are p-isomorphic. (Here a one-to-one function $f$ is poly-time invertible if there is a poly-time function $g$ such that $g(f(x)) = x$ for all $x$, and $g(y)=\bot$ if $y$ is not in the image of $f$.) Two languages being p-isomorphic basically just means that one is a re-encoding of the other, and thus preserves the "structure" of the other in some sense, that I think is similar to what you're talking about. All known natural $\mathsf{NP}$-complete problems are p-isomorphic. The idea of the proof is a ping-pong-type argument.

Finally, as another pointer to the literature: Although the notion of $\mathsf{NP}$-complete is not sensitive to the size of the reduction (so long as it's bounded by a polynomial), the Exponential Time Hypothesis (ETH) and its variants are sensitive to super-linear blow-ups in size. So there may be some work on ETH that addresses parts of your question.