I'm curious to learn more.

My general impression (having studied computer science, and complexity theory in passing, but being far from an expert) is that bounded-error approximations are not as well studied as precise solutions. I also get the impression that most of the bounded-error complexity results are results for specific approximation algorithms, not proofs about the limits of such algorithms. Are impossibility results for bounded-error approximations common?

In other words,

the set of NP-complete problems where there is a polynomial-time approximation scheme is not that large

Do you mean the set where such approximations are known, or the set where they are not known to be impossible?