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Your goal is to build such a ${V_\alpha}$. The open cover $U = \cup_p U(p)$ has a locally finite refinement $U = \cup_{i \in I} V_i$. This means that for each $i \in I$, $V_i \subseteq U(p_i)$ for some $p_i$. We might as well assume (by renaming $p_i$) that $p_i \in V_i$, and we can now rename the $V_i$ to $U(p_i)$. So without loss of generality, your cover $\{U(p)\}$ has a locally finite subcover $\cup_i U(p_i)$. Now your goal is to use this local finiteness property to find the $V_\alpha$. This seems to be the importance of the paracompactness condition, but I'll let you finish the proof.
For my money, a function with essential singularities seems "pathological". Functions that are meromorphic on the full extended complex plane are basically algebraic objects with all sorts of wonderful rigidity. On the other hand, the big Picard theorem says that if a function has an essential singularity at $z$, the range of $f$ on any neighborhood of $z$ misses at most one point from $\mathbb{C}$! These beasts also lie behind holomorphic functions on e.g. the disk which can't be extended to bigger domains.
Note that it's even true that an artin local ring $k$ which is reduced is equal to its residue field. This is because the unique prime ideal of an artin local ring is its nilradical (the nilradical is always the intersection of all prime ideals). It's also equivalent to the other statement since reduced = regular in codimension $0$.
Paracompactness implies that there is a locally finite cover $U = \cup_{p_i \mid i \in I} U(p_i)$ (my $U(p_i)$ may be smaller than your $U(p_i)$). This means that each $U(p_i)$ only intersects finitely many $U(p_j)$. I think it is not likely that $s(p)|_{U(p) \cap U(q)} = s(q)|_{U(p) \cap U(q)}$, but you should be able to use the local finiteness to select a single small enough cover where the intersections actually match. (it's not immediately obvious to me how to make this argument).
thanks! I'd be happy to accept either of these comments as an answer. (I was thinking in terms of coordinates in projective space and missed the adelic interpretation).
I'm not sure about being a subquotient of a torsion divisible group (perhaps one can always embed a torsion group into a torsion injective, hence divisible group?), but $M$ torsion divisible does not imply $H^i_{\mathrm{cont}}(G, M)$ is torsion divisible. For example, take $G$ as the absolute Galois group of $\mathbf{F}_p$ and $M = \overline{\mathbf{F}_p}^\times$. This is torsion divisible since every element algebraic over $\mathbf{F}_p$ is a root of unity but $\overline{\mathbf{F}_p}$ is algebraically closed. Then $H^0_{\mathrm{cont}}(G, M) = \mathbf{F}_p^\times$, which is not divisible.
I would not expect this to be true for the latter property: let $G$ be a free pro-p group on infinitely many generators, and let $M$ be an abelian pro-p group with trivial $G$-action (e.g. $M = \mathbf{Z}[p^{-1}]/\mathbf{Z}$. Then $H^1_{\mathrm{cont}}(G, M) = \mathrm{Hom}(G, M)$, and the universal property of free pro-p groups should imply that this is infinitely generated. Of course, subquotients of finitely generated abelian groups are finitely generated, so no help by relaxing to that.
@nfdc23's comment addresses the question in the cases I'm looking for. I would accept this as an answer or another answer which addresses the generality I asked about in the question.
@nfdc23 Thanks, this clears up how to get the $\mathrm{Ext}$ construction to work more generally. Are all cohomologies one should care about of the form $H^0(F) = \mathrm{Hom}(O, F)$ where $O$ is the tensor unit in the category (it seems like you used this to get the map from $\mathrm{Ext}^j(O, G)$ to $\mathrm{Ext}^j(F, F \otimes^L G)$, right?) ?
@DenisNardin I'm happy working in the derived category, but there's a big jump from this to dg categories, etc., isn't there? If such language is necessary to discuss this, that's fine, but I'd like to see a result formulated similarly to the one in the question deduced as a consequence of whatever is proved in a fancier setting.
The L-(M-B) book has a serious error, and does not address some of the fundamental theorems of etale cohomology which are proved via Chow's lemma (e.g. proper base change). Look at Olsson's paper "Sheaves on Artin Stacks" https://math.berkeley.edu/~molsson/qcohrevised.pdf for an updated treatment that fixes this error, treats some more recent developments, and moreover is in English.
Why do you say "finite étale" extensions of $K_1, K_2$ as opposed to "finite separable" extensions? Is there a general statement that holds for bases other than fields?