I have studied the proof of the (downward) Lowenheim-Skolem theorem - as an undergraduate, so you should take my conclusions with some salt - but my understanding of the (downward) Lowenheim-Skolem theorem was exactly that the proof builds a model out of the syntax of the first-order theory in question.

I'm not saying that the proof is trivial - what I'm saying is that holding Godel-numberability and the possibility of a strict formalist interpretation of mathematics in your mind provides a helpful intuition for the result.