You seem to already have answered your question when you realized that in order to deduce the sentence $Con(ZFC)$ using Gödel's completeness theorem, you need to have a set model existence of which is impossible to be proven by the incompleteness theorem.
If you believe that there exists a Platonic universe of sets where certain objects exist and satisfy axioms of ZFC, then you believe that ZFC is consistent because there is a "model" of it. On the other hand, if you try to formalize this belief within ZFC and want to prove that ZFC is (formally) consistent, then you need to have a set model.
Let me give another example which may clarify things for you. As you might already know, it is a theorem that L is a model of ZFC. But what does this theorem really mean, how do we express it in ZFC? Since L is a proper class, you cannot use the formalized truth predicate (which is defined only for sets) to say each axiom of ZFC is true in L.
Instead, what you do when you want to claim that a definable proper class M is a model of ZFC is that you relativize your quantifiers to this proper class and show that each axiom of ZFC is true, i.e. you want to prove $\phi^M$ for all axioms $\phi$.
From this perspective, what does it mean to say "V is a model of ZFC"? V is the universe, which you can define by $\phi(x): x=x$. Then the equivalence $\phi^V \leftrightarrow \phi$ is trivially true for all sentences, right?
The issue is expressing "is a model of ZFC" part in a sensible way for proper classes. As Joel suggested, you can work with stronger set theories.