If $L/K$ is a Galois extension, then one can define a site in which the objects are intermediate fields $K \subseteq E \subseteq L$ which are finite over $K$. Or, you can take finite étale algebras $A$ over $K$ with a homorphism of $K$-algebras $A \to L$. The topology is the étale topology. These two sites define the same topos. There is an equivalence between sheaves on either of these sites and discrete $\mathrm{Gal}(L/K)$-modules, hence cohomology coincides with Galois cohomology.
Does this answer your question? I am not sure I understand what you have in mind.
By usual (sometimes not so trivial) homological arguments, one can reduce to the case where $M$ is a finite discrete module over an artinian ring of residual characteristic $p$. In that case, I think you want $S$ to contain places above $p$ as well, even if your $M$ is unramified at $p$, so let me assume this.
The module $M$ induces an étale sheaf $M_{et}$ on $\operatorname{Spec}\mathcal O_{L,S}$ for all finite extension $L/K$. The spectral sequence UPDATE (converging to $H^{i+j}(\operatorname{Spec}\mathcal O_{K,S},M_{et})$)
$$E_{2}^{i,j}=\underset{\longrightarrow}{\operatorname{\lim}}\ H^{i}(\operatorname{Gal}(L/K),H^{j}(\operatorname{Spec}\mathcal O_{L,S},M_{et}))$$
then induces isomorphisms between $E_{2}^{i,0}$ and $H^{i}(\operatorname{Spec}\mathcal O_{L,S},M_{et})$ or in other words $H^{i}(G_{K,S},M)$ is isomorphic to $H^{i}(\operatorname{Spec}\mathcal O_{K,S},M_{et})$. So you can assume that you are working with Galois cohomology throughout $provided$ you use Galois cohomology with restricted ramification.
Because the Tamagawa Number Conjectures are formulated only in the setting above, Bloch and Kato could have used Galois cohomology instead of étale cohomology everywhere without changing anything. To touch upon your last question, I think there are two reasons why they chose étale cohomology.
First, at least at the time they wrote, Galois cohomology was not the most familiar object of the two. In fact, many classical well-known results were given correct complete proofs only very late (in the late 90s in some cases). On the other hand, SGA (and works of Bloch and Kato themselves) existed as references for étale cohomology.
Second, using étale cohmology, one can formulate the TNC over more general bases than $\operatorname{Spec}\mathcal O_{K,S}$ (for instance any scheme of finite type of $\mathbb Z[1/p]$). This kind of generalization had been the key idea of previous works of Kato and Bloch-Kato on higher class field theory so it is not surprising that they decided to at least allow the same kind of generality in their subsequent works.
Best Answer
All Galois cohomology groups are given by etale cohomology, but you seem to be asking the opposite. For an etale sheaf $F$ on $X$, there is a Hochschild-Serre spectral sequence $H^{p}(\pi_{1}(X),H^{q}(\tilde{X},F))\implies H^{p+q}(X,F)$ where $\tilde{X}$ is the "universal covering scheme" of $X$. Don't expect this to collapse to give isomorphisms $H^{p}(\pi _{1}(X),F)\approx H^{p}(X,F)$ except when $X$ is spec of a field or a henselian local ring. There are a few fragmentary results. For example, $H^{1}(\pi _{1}(X),A)\approx H^{1}(X,A)$ when $X$ is an open subscheme of the spec of the ring of integers in a number field and $A$ is an abelian scheme on $X$.