I don't see how Gödel's theorems at all shows that "the introduction of the idea of the truth erodes the truth". Gödel's incompleteness theorems are mostly about provability anyway, which is a distinct concept from truth.

Though, Gödel's completeness theorem I suppose does relate provability to truth in that it shows that (for systems with the right kind of rules of inference) provability is equivalent to something being true in all models...

Still, are you sure Tarski's undefinability theorem isn't more relevant to your point?