As someone who initially thought of the completeness theorem (= Godel's observation that first-order logic has a 'good proof system') as just an annoying digression from the really important semantics-focused model theory I was interested in, I think it helps to consider how the existence of a good proof system drastically improves complexity bounds. (See also the discussion here about whether/why the completeness theorem should be "expected.")
Suppose I have a logic $\mathcal{L}$ where the entailment relation $\models_\mathcal{L}$ is defined in some set-theoretic way - that is, $\mathfrak{M}\models_\mathcal{L}\Gamma$ means "There is some set-theoretic configuration 'built on' $\mathfrak{M}$ such that [properties]." In the case $\mathcal{L}=\mathsf{FOL}$, this set-theoretic configuration is just a collection of Skolem functions (or similar) witnessing the truth of the relevant sentences. The point is that at first glance the induced entailment relation on (sets of) sentences is $\Pi_1$ in the Levy hierarchy: "$\Gamma\models_\mathcal{L}\varphi$" means "There is no structure-plus-configuration pair $(\mathfrak{M}, \mathbb{X})$ such that $\mathbb{X}$ witnesses the truth in $\mathfrak{M}$ of $\Gamma\cup\{\neg\varphi\}$." The existence of a good proof system brings things all the way down to $\Sigma_1^{0}$ in the arithmetical hierarchy, since we can replace "There is no countermodel" with "There is a proof." This is a gulf in complexity that is difficult to appropriately summarize; suffice to say that it's quite impressive! Even if you don't care about decidability issues, the indicates a genuine reduction in the ontological commitments needed to make sense of the surrounding questions. (It also forms a nice contrast with the situation with respect to Herbrand semantics; in Herbrand semantics, the individual elements considered are much simpler but the entailment relation induced is much more complicated. See here.)
Admittedly, it is absolutely possible to do lots of research in model theory using just the compactness theorem. Think of the compactness theorem as saying that a good proof system exists for $\mathsf{FOL}$ except maybe it might be really computationally complicated. It turns out that for most pure model-theoretic questions, we really only care about the "finite based"-ness of $\models_{\mathsf{FOL}}$, that is, the fact that $\Gamma\models_\mathsf{FOL}\varphi$ iff $\Gamma_0\models_{\mathsf{FOL}}\varphi$ for some finite $\Gamma_0\subseteq \Gamma$. I have two responses to this:
There are times that you do care about the exact complexity of $\models_\mathsf{FOL}$, and for this an explicit proof system is really the best way to package all the necessary information. (For example, look at the arithmetized completeness theorem in the context of model theory of theories of arithmetic, like first-order Peano arithmetic.)
We should always be looking for ways to connect seemingly-disparate perspectives on any piece of mathematics, and the broad topic of "structures, theories, and properties" is no different. I would provocatively, but not wholly jokingly, say that the real value of the completeness theorem to the model theorist (over compactness as such) is that it helps develop connections with proof theory.
