Asking about non-standard models of ZFC is deeply connected to asking about ZFC with other axioms added. This is connected to the Löwenheim–Skolem theorem and related results. Note for example that if there is some large cardinal axiom L and statement S such that ZFC + L can model ZFC + S, and L is independent of ZFC, then ZFC + S is consistent if ZFC is.

For example, just what do we mean in talking about a 'model' of ZFC, when ZFC or something similar is exactly the raw material used to construct models in other fields?

We can make this precise by talking about any given set theory as your ground and then discussing the models in it. This is connected to Paul Cohen's work in forcing but I don't know anything about it in any detail. The upshot though is that we can talk about models in helpful ways.

But they are not particularly useful to most of the people who find them interesting. Most of the applications that Barwise suggests for Aczel's work can be modeled with just a little more effort in standard ZF or ZFC. And I just can't imagine that an AI researcher will learn anything from the p-zombie debate which will tell him which features or mechanisms his AI must have so as to avoid the curse of zombiedom.

Not much disagreement there, but I think you might underestimate the helpfulness of thinking about different base axioms rather than talking about things in ZFC. In any event, the objection is not to your characterization of thinking about p-zombie but rather the analogy. The central point you are making seems correct to me.