He's right, the Godel theorems have nothing to do with existence of models satisfying this-or-that. Such would be "semantic" truths. The reason Hilbert's program survived Löwenheim–Skolem is that Hilbert was a formalist concerned with "syntactic" truth, that is, whether there are statements P such that neither P nor not-P could be proven by the axiom system.

You might think LS would trivially demonstrate as much---"Just take P = our underlying model is countable!"---but this is not formalizable within the system itself.