Lately I've started looking into approximation algorithms for NP-hard problems and I was wondering about the theoretical reasons for studying them. (The question is not meant to be inflammatory - I'm merely curious).
Some truly beautiful theory has come out of the study of approximation algorithms - the connection between the PCP theorem and hardness of approximation, the UGC conjecture, the Goeman-Williamson approximation algorithm, etc.
I was wondering though about the point of studying approximation algorithms for problems like Traveling Salesman, Asymmetric Traveling Salesman and other variants, various problems in mechanism design (for instance in combinatorial auctions), etc. Have such approximation algorithms been useful in other parts of theory in the past or are they studied purely for their own sake?
Note: I'm not asking about any practical applications since as far as I know, in the real world, it's mostly heuristics that are applied rather than approximation algorithms and the heuristics are rarely informed by any insight gained by studying the approximation algorithms for the problem.