All Serre needed to use was quasicoherent sheaves. For these it's a fundamental theorem that Cech cohomology is the "correct" cohomology. That means that it agrees with the cohomology defined abstractly, via derived functors. Since derived functors give rise to long exact sequences, the fundamental theorem implies that short exact sequences of (quasi)coherent sheaves yield long exact sequences in Cech cohomology. This just isn't true for coarser abelian categories like arbitrary sheaves or presheaves.
For an idea of why the theorem is true for quasicoherent sheaves, it's generally true that Cech cohomology becomes the right cohomology when the space has a "good cover," which in this case means a cover whose elements have no higher cohomology for whatever sheaf we're investigating. This is always true for quasicoherent sheaves by a reduction to affine varieties and then commutative algebra, which isn't possible for general sheaves since there may be no cover on which the sheaf is that associated to a module over a ring.
Let $F$ be a sheaf of Abelian Groups on the Zariski site. Let us extend the definition as described in the question of the Zariski sheaf $F$ to a functor defined on the etale site and call it $F_{et}$. We first verify that this is indeed a sheaf.
First certain preliminaries.
Let $g : U \rightarrow X$ be an etale open set in $X$, then $F_{et}(g:U \rightarrow X) = F(g(U))$. This is easy to see by the definition of the pullback of a sheaf and the fact that $g(U) \subset X$ is an open subset in the zariski topology, since $g$ is etale in particular flat. Next, let us see what happens on the intersection of two such open sets.
Let $g : U \rightarrow X$ and $h : V \rightarrow X$ be two open subsets. Let $g\times_X h : U \times_X V \rightarrow X$ be the fiber product. Then $F_{et}(g \times_X h : U \times_X V \rightarrow X) = F((g \times_X h)(U \times_X V \rightarrow X)) = F(g(U) \cap h(V))$. The first equality follows as above while the second equality follows from the fact that $(g \times_X h)(U \times_X V \rightarrow X) = g(U) \cap h(V)$. This is explained below :
We have the following setup $U \rightarrow g(U) \hookrightarrow X$, where the first arrow is surjective and the second is an open immersion(topologically and if we give (and we do) $g(U)$ the induced subscheme structure, then scheme theoretically). A similar factorization exists for the arrow $h$. Using the universal property of cartesian diagrams we get that $g \times_X h$ factors through $g(U) \times_X h(U) = g(U) \cap h(V)$(This is clear since $g(U)$ and $h(V)$ are subsets of $X$). Let us analyze the map $U \times_X V \rightarrow g(U) \times_X h(U)$.
Claim : The above map $U \times_X V \rightarrow g(U) \times_X h(U)$ is surjective.
Proof of Claim : Note that this map factors as follows $U \times_X V \rightarrow g(U)\times_X V \rightarrow g(U) \times_X h(U)$. Now, note that $U \rightarrow g(U)$ and $V \rightarrow h(V)$ is surjective. We know that base change of a surjective morphism is surjective. Hence both the arrows are surjective maps and hence the composition is surjective. Thus proving the claim.
Now, we prove that $F_{et}$ is infact a sheaf. Let $\lbrace g_i : U_i \rightarrow U \rbrace$ be an etale cover of an etale open set $g : U \rightarrow X$ i.e. each of the $g_i$ is an etale map and $\cup_i g_i(U_i) = g(U)$. We consider the standard sequence of Abelian groups
$F_{et}(g:U \rightarrow X) \rightarrow \prod^i F_{et}(g_i) \rightarrow \prod F_{et}(g_i \times_X g_j)$
From above paragraphs, the exact sequence is same as the exact sequence
$F(g(U)) \rightarrow \prod F(g_i(U_i)) \rightarrow F(g_i(U_i) \cap g_j(U_j))$
This is an exact sequence follows from the fact that $F$ is a sheaf on the zariski site.
Thus we have that $F_{et}$ is infact a sheaf. Note that the etale cohomology of this sheaf will be the same as usual cohomology as can be seen by considering cech complexes.
This is infact functorial for the morphism for sheaves.
Best Answer
Studying coherent sheaves on the small étale site of a scheme $S$ is not tremendously interesting: according to Proposition 35.8.9 [03DX] of The Stacks Project, there is a natural functor which gives an equivalence of categories between the category of quasi-coherent sheaves on $S$ and the category of quasi-coherent $\mathcal{O}$-modules on the small étale site of $S$.
On the other hand, once you go deeper down the rabbit hole and start considering sheaves on stacks, the étale site reappears. For example, one can define quasi-coherent sheaves on a Deligne-Mumford stack by descending quasi coherent sheaves from its étale presentation by a scheme.