While more generality is not always better, we often find it easier to prove a more general result. For instance, although one can prove the unsolvability of the quintic without invoking Galois theory explicitly, that's surely the wrong way to do it: the right way is to develop Galois theory along the way, since it's the natural narrative the problem lives in.
Similarly, model theory is one way of moving from specific settings (e.g. the theory of groups) to more general ones (arbitrary theories). It is certainly not the only way - category theory is a wildly different (and in terms of applications, more successful) example - but it is one way, and it should be no more surprising that it has applications than that any generalization does. In particular, your objections ("But what's the significance of doing that? Would this be an exercise in futility, since we would have to fall back to algebraic reasonings anyway?") could be made to any approach to generalizing anything.
So what are some applications of model theory?
Well, let me begin by addressing an error in the OP and the comments. Theories define classes of structures - e.g. the group axioms constitute a theory, and a model of the group axioms is exactly . . . a group! There are non-isomorphic groups, as I'm sure you're aware. Theories do not in general determine structures up to isomorphism. A version of this (even for complete theories!) can be made precise and proved via the compactness theorem, so your claim that all models of a theory $T$ are "the same" is completely false. There is in general no good sense in which any two models of a theory $T$ are equivalent. (There is an equivalence relation coming from model theory, elementary equivalence, which applies when two structures satisfy all the same first-order sentences - or equivalently each satisfy some complete theory $S$ - but this only makes sense in the context of complete theories, not theories in general.)
Interestingly, compactness turns out to be a useful tool for developing applications! A good example is the Ax-Kochen theorem. Another application is the existence of the hyperreals, a kind of ordered field which allows calculus with infinitesimals to be formally developed; incidentally, compactness here is specifically applied to a complete theory, so it's even valuable to move between two elementarily equivalent structures! And for a much more advanced example, consider the role of compactness for proving transfer principles in motivic integration.
You also dismiss the use of definable elements and sets. This is a mistake. By analyzing the structure of definable (and definable-with-parameters) sets in a structure, we can prove results about the structure itself. O-minimality has been especially useful in this regard.
Finally, connecting back to Galois theory, model theory provides general approaches to Galois-like theories, and was particularly useful in resolving a number of questions in differential Galois theory.
Let me add one more (and then I'll stop since I think I've made my point). A very logic-y theorem is the (downward) Lowenheim-Skolem theorem; roughly speaking, given a "big" structure we may find a "small" substructure satisfying all the same sentences.
Even this result, which is couched specifically in terms of first-order sentences, has applications! There are several examples in general topology that I'm aware of; see e.g. this paper (but there are many others).
You have the right impression: most applications of model theory to algebraic geometry are essentially "classical", in the sense that they are about varieties as definable sets over algebraically closed fields, not (explicitly) about sheaves and schemes. The model theoretic approach is actually very similar to the view of the foundations of algebraic geometry promoted by André Weil in the 40s (the monster model is essentially Weil's universal domain) which was largely superseded by Grothendieck's approach not long after.
One obstruction to the use of sheaves in model-theoretic algebraic geometry is that it is the constructible topology, not the Zariski topology, which is most natural in model theory. Here algebraic sets are clopen, not just closed - this corresponds to the fact that first-order languages are closed under complement (negation) and projection (quantification). Of course, the constructible topology is totally disconnected, which removes much of the geometric content captured by sheaves.
On the other hand, model theory has proven to be very useful in fields adjacent to algebraic geometry, where a good theory of schemes is not available or is much more complicated than in the classical case. I'm thinking of semialgebraic geometry (and o-minimal generalizations), differential algebra, difference algebra, Berkovich spaces, etc.
That's not to say that schemes and sheaves are nowhere to be found in model theory - I'll leave it to someone else to give some references to some places where they appear. For my part, I'll link you to Angus Macintyre's paper Model Theory: Geometrical and Set-Theoretic Aspects and Prospects from 2003. Here Macintyre surveys the history of model theory and suggests (in a somewhat vague way) a future in which model theory has more in common with Grothendieck-style algebraic geometry. I think this paper has been fairly influential, but the revolution hasn't arrived yet.
Best Answer
If by applications you allow applications to other mathematical field, then here two examples.
The first one is Ax's theorem. It states that a polynomial function $\mathbb C^n \to \mathbb C^n$ is injective if and only if it is bijective. Basically, the proof is as follow : it is clearly true for (polynomial) functions $\mathbb F^n \to \mathbb F^n$ for any finite field $\mathbb F$ ; the Nullstellenstatz then implies that is also true for polynomial functions $\overline{\mathbb F}^n \to \overline{\mathbb F}^n$ with $\overline{\mathbb F}$ the algebraic closure of a finite field ; then comes the model-theoretic argument with Los's theorem which shows that it is again true in the ultraproduct of all algebraic closure of finite field. Then it can be seen that this ultraproduct is an algebraically closed field with characteristic $0$ and infinite transcendence degree over $\mathbb Q$, that is $\mathbb C$.
The second example is the Nullstellensatz itself. It is an immediat corollary of the model-completeness of the theory of algebraically closed fields.