Hi David. I guess the answer depends on what one means by "purely model theoretic". The closer to such a thing that I know of is an argument of Woodin. It is similar to an argument of Jech, which though mathematically alike, is more "proof theoretic" in nature. I wrote some notes on it. They are here.

The only ingredient that is not strictly model theoretic is the fixed point lemma.

Hi David. The following is ("probably", he says) not what you want, but I am leaving it here, as it may explain some of the context. It is an argument of Woodin, similar to one of Jech, which though mathematically alike, is more "proof theoretic" in nature. I wrote some notes on it; they are here.

The key fact you refer to is however not proved model theoretically, but by an appeal to the arithmetic fixed point lemma, also a key fact in Gödel’s approach.

The idea is the following (the note provide the missing details and makes this precise): A property $P$ of models of set theory is hereditary iff whenever $P(M)$ and $N\in M$ is a model of set theory such that $M$ thinks that $P(N)$, then in fact $P(N)$.

Then the following holds:

For any hereditary $P$, either $P(N)$ fails for all $N$, or else there is an $M$ such that $P(M)$ but $P(N)$ fails for all $N\in M$.

This is proved by an application of the fixed point lemma: Take $\phi$ such that (ZFC proves tha) $\phi$ holds iff for all $N$, $P(N)$ implies $N\models\lnot\phi$.

It is immediate that if $P(M)$ holds for some $M$, then it holds for some $M$ such that $M\models\phi$. But then $M$ thinks that there is no $N$ satisfying $P$.

Ok. If $P(N)$ is "$N$ is a model of set theory", then $P$ is hereditary, and we have the second incompleteness theorem. As shown in the note, quite a few similar results follow all at once by considering appropriate properties $P$.