[ttctciyf]

Isn't the claimed "truth" based on an appeal to a more intuitive notion of truth than a formal model-relative one?

The Godel sentence for some Formal Axiomatic System asserts, by way of numerical encoding, its own unprovability in that FAS, and since it in fact cannot be proven in the FAS it is "true" in the sense that it asserts (via its numerical encoding) something which is in fact the case, i.e. just the ordinary sense of "true".