Is it ever a good idea to use Keisler-Shelah to show elementary equivalence?

People in algebra use Keisler-Shelah a lot. For example, Malcev proved that for fields $F,K$ one has $GL_m(F)$ is elementary equivalent to $GL_n(K)$ iff $m=n$ and $F$ is elementarily equivalent to $K$. The idea is first you prove that this is equivalent to $M_m(F)$ is elementarily equivalent to $M_n(K)$. Then you take ultrapowers and use that taking matrices commutes with ultrapowers to deduce that $M_m(F)$ and $M_n(K)$ have isomorphic ultrapowers iff $m=n$ and $F,K$ have isomorphic ultrapowers, i.e. $F,K$ are elementarily equivalent.

There are other examples like this in algebra. A good reference is the paper On Malcev's theorem on elementary equivalence of linear groups by Beidar and Mikhalev (Contemp. Math. 131 Part I (1992), 29-35).

I believe Sohrabi and Miasnikov also use Keisler-Shelah in their work on elementary equivalence of nilpotent groups.

Here is an instance where it seems fine to use Keisler-Shelah.

Let ${\cal A}$ be the graph consisting of a single infinite beaded chain, or more concretely, the integers under adjacency. That is, ${\cal A}=\langle\mathbb{Z},\sim\rangle$, where $n\sim m$ just in case they differ by exactly one.

And let ${\cal B}$ consist of two (or more) disconnected copies of ${\cal A}$.

It is easy to see that the ultrapower of either ${\cal A}$ or ${\cal B}$ by any ultrafilter on a countable index set consists of continuum many such beaded chains. Thus, the structures ${\cal A}$ and ${\cal B}$ have isomorphic ultrapowers, and so they are elementary equivalent by Keisler-Shelah.

(Essentially the same argument came up in my answer to Stefan Geschke's queston Is non-connectedness of graphs first order axiomatizable?, but I had used Löwenheim-Skolem there rather than Keisler-Shelah, so this example may not satisfy run afoul of your requirements. But I think the argument from Keisler-Shelah is no harder or easier than the argument from Löwenheim-Skolem.)

As in Benjamin Steinberg’s answer, these are examples related to algebra.

The Ax–Kochen–Ershov principle states that two unramified henselian valued fields are elementarily equivalent iff their value groups and their residue fields are respectively elementarily equivalent. (There are many variations and extensions of the statement in the literature.) An outline of the proof is as follows: we may assume the two fields are countable and that CH holds. We take ultrapowers over a uniform ultrafilter on $\omega$: the value groups and residue fields of the ultrapowers are then isomorphic, and one can use this and the saturation of the ultrapowers to construct a valued field isomorphism by a back-and-forth transfinite sequence of countable partial isomorphisms.

Likewise, a crucial part in Ax’s analysis of the theory of finite and pseudofinite fields is the result that two pseudofinite fields with the same absolute numbers (i.e., the relative algebraic closure of the prime field) are elementarily equivalent. Again, this is shown by a back-and-forth construction of an isomorphism of their ultrapowers.