Since 2 NP-complete problems are by definition reducible to each other, so a solution to one of them can be obtained by using a black-box solving the other one, why don't they have similar approximation ratios (referring to their optimization counterparts)? I guess that some constant or even polynomial drift might be understood but we have the case of constant-factor approximation algorithms for some NP-complete problems and, on the other hand, other problems that cannot be even approximated by a polynomial-ratio approximation algorithm, such as general TSP? Thank you
Reductions are defined with respect to the decision version of the problems. Approximation ratios for their optimization versions are a separate question, which seems related but doesn't necessarily have to be. So to answer your question with a question, from a philosophical perspective, why should you expect the class NPC to preserve approximation ratios when it isn't defined with respect to them in the first place?