Good catch! I was thinking of the Coates-Wiles theorem, which I'd forgotten only applies in certain ranks. Is it now true that the best results known in the CM case are the same as in the general case?

I just stumbled on this old question, and it might be useful to note that a very nice expository book on the subject has appeared in the meantime: Lectures on Formal and Rigid Geometry by Bosch. Some of the heavy lifting is punted to the BGR book (same first author!), but this book is much easier to read and explains a lot of the motivation and geometric intuition of the subject.

Perhaps meditating on the discussion of continuous group cohomology in Scholze's paper as well as Exercise 14 here might shed some light into what's going on here. swc.math.arizona.edu/aws/2017/2017BhattNotes.pdf

BTW, the base change theorems are the basic reason why good reduction implies unramified Galois action (the converse is harder of course); compatibility of the base change isomorphisms with Galois shows us that inertia (the kernel of the restriction map on Galois groups) acts trivially.

Now, the trace of the Frobenius automorphism generating $\overline{k(s)}/k(s)$ on $H^i(\mathcal{X}_{\overline{k(x)}}, \underline{\mathbf{Q}_\ell)}$ is related to $|\mathcal{X}(s)|$ by the Grothendieck-Lefschetz trace formula. We can lift this to an element of $G_K$ (think of the case where $S = \mathrm{Spec} \mathscr{O}_K$), and the isomorphism coming from the base change theorems respects traces of elements of the Galois group, so we win.

The result of Serre seems to just be a consequence of the Grothendieck-Lefschetz trace formula plus some base change theorems. For simplicity, let's say that $X/K$ is $K$-smooth (you can probably avoid this using stronger results, e.g. Deligne's Weil II). Then given some $\mathcal{X}/S$, we can shrink $S$ so that we can assume without loss of generality that $\mathcal{X}/S$ is smooth. Then the smooth and proper base change theorems say that we can identify $H^i(X_{K^s}, \underline{\mathbf{Q}_\ell})$ with $H^i_c(\mathcal{X}_{\overline{k(x)}}, \underline{\mathbf{Q}_\ell)$.

Do you need $\ell$ not to divide the order of the isotropy groups for this? Is it true that the (geometric) fibers of $\pi$ are classifying spaces for the stabilizer groups?

Ah great! This is what I was hoping for - does anyone know what conditions on $\mathscr X$ make $\mathscr X_x = B \mathrm{Stab}_x$? Also, does proper base change work when $\ell \mid |G|$ (but isn't equal to the field characteristic)? I'd love to read a proof of these sorts of basic etale cohomology theorems which work for stacks.

@DanPetersen Can you give a reference/argument for the fact that $H^i_{\mathrm{et}}([X/G], \mathbf{Q}_\ell) \simeq H^i_{\mathrm{et}}(X/G, \mathbf{Q}_\ell)$ when $\ell \nmid |G|$? What hypotheses on $X$ are used for this fact? (I like the argument you give in the answer very much, but was curious about the general phenomenon)

Since this question popped up again, I'm surprised nobody has mentioned the excellent "History of Class Field Theory" notes by Keith Conrad. It's fairly short and very clearly discusses the various approaches and formulations of class field theory and their relationships, and provides good pointers to original sources.