I think most people who know Goedel's theorem say "G is true" and are "unreflective platonists," by which I mean that they act like the natural numbers really exist, etc, but if you pushed them on it, they'd admit the doubt of your last couple of sentences.

Similarly, most people (eg, everyone on this thread), state Goedel's completeness theorem platonically: a statement is provable if it is true in every model. That doesn't make sense without models having some platonic existence. (yes, you can talk about internal models, but people don't.) I suppose you could take the platonic position that all models exist without believing that it is possible to single out the special model. (Eliezer referred to "the minimal model"; does that work?)