I have seen the model existence theorem many times used to show the consistency of a theory. It works as follows. Let $T$ be a theory. A model $\mathcal{M}$ is given for $T$ and therefore, by the model existence theorem, $T$ is thereby known to be consistent. My question is, does anyone have specific examples of the reverse direction used to prove that a theory is incomplete? Specifically, I am looking for examples in the literature where it was unknown if $T$ was consistent or not.
That is, for a specifc theory $T$, first show directly that there are no models for $T$, and as a result conclude that $T$ is inconsistent.
I think that nearly every proof of the inconsistency of a theory works the way you are describing. It's very rare to see someone actually exhibit a sequence of formal sentences built according to the rules of a formal proof system and that ends in $s \wedge \neg s$ for some statement $s$.
You mentioned Kunen's proof of the inconsistency of the theory $T = $ ZFC+"there is a non-trivial elementary embedding from $V$ to $V$". A formal syntactic proof of this would be a nightmare. Instead, Kunen argues that there are no models of the theory (the proof begins by assuming that you have an elementary embedding from $V$ to $V$, where $V \models ZFC$, that is, it starts with a model of $T$). When he reaches a contradiction he concludes that the assumed elementary embedding did not exist, thus showing that there are no models of $T$. It's still a proof by contradiction, as non-existence proofs often are, but it's very far from being a syntactic derivation of $s \wedge \neg s$ in a formal proof system.