You're correct, but I think the spirit of what GP wrote is still true. You can't capture an infinite structure fully by "finitary" methods, either you use FOL

You're correct, but I think the spirit of what GP wrote is still true. You can't capture an infinite structure fully by "finitary" methods, either you use FOL and then you run into L-S, or you use higher-order logic (for which L-S doesn't apply) but then you don't have a complete proof system anymore. To tie it all together, L-S and incompleteness are about different flavours of "not being able to capture something". L-S is about models of different cardinalities. These models do still all satisfy exactly the same sentences. Incompleteness is about different models actually satisfying different sentences.