Is there a conceptual reason why the notion of "quasicoherent sheaf" is independent of the choice of topology? Let $X$ be a scheme and $\mathcal S$ a site which is a full subcategory of the category $Aff/X$ of affine schemes with a map to $X$. If I understand correctly, the category $QCoh^\mathcal S(X)$ of $\mathcal S$-quasicoherent sheaves is the global sections of the $\mathcal S$-stackification of the functor $Mod: \mathcal S^{op} \to Cat$, $(Spec A \to X) \mapsto Mod_A$, where $Mod_A$ is the category of $A$-modules.
The interesting phenomenon (cf. the stacks project, which is probably using slightly different definitions) is that for reasonable $\mathcal S$ (the above link gives precise conditions), $QCoh^\mathcal S(X)$ is actually independent of $\mathcal S$. I'm looking for a high-concept explanation of this fact.
The best I can figure is the following. For reasonable topologies, the functor $Mod: \mathcal S \to Cat$ is already a stack, so its global sections can be computed using an $\mathcal S$-cover of the terminal object (which I've technically left out of the category $\mathcal S$, but that's okay). Since $X$ is a scheme, it comes with a Zariski cover, which is also a $\mathcal S$-cover for reasonable topologies. Thus $QCoh^\mathcal S(X)$ is simply computed in the same way for reasonable topologies $\mathcal S$, so of course it agrees.
This suggests that a statement of this meta-principle (an alternative to the one found in the stacks project) would say that 

Claim: Let $X$ be a scheme $\mathcal S$ be a full subcategory of $Aff/X$, equipped with a Grothendieck topology. Then $QCoh^\mathcal S(X) = QCoh^{Zariski}(X)$ if
  
  
*
  
*$Mod : \mathcal S^{op} \to Cat$ is a stack, and
  
*Every Zariski cover is an $\mathcal S$-cover.

Questions:


*

*Is the above claim correct?

*If not (or if so!) is there some other high-concept way to see that $QCoh^\mathcal S(X)$ is independent of $\mathcal S$ for reasonable $\mathcal S$?
 A: The basic reason is that a sheaf of modules is quasi-coherent if and only if it is a Cartesian presheaf. Therefore a certain covering suffices for the condition to hold. The condition on coarse covers determine the behavior on fine covers, therefore being quasi-coherent in Zariski topology (or in étale topology on an algebraic space, DM-stack) already makes the sheaf quasi-coherent in any finer topology. The theory is specially simple when you have a base of "affine" schemes that cover the base space.
The Cartesian condition is just a condition on presheaves, so, in a sense, topology is not important. Risking to commit self-promotion, this is treated in the context of geometric stacks in the paper "A functorial formalism for quasi-coherent sheaves on a geometric stack", https://arxiv.org/abs/1304.2520 (see also Expo. Math., 33 (2015) pp. 452-501). The case of affine schemes is explained in section 2 and it is generalized to stacks with affine diagonal in section 3. The case of a semi-separated scheme follows adapting (and somehow simplifying) the arguments there. In particular, one does not need to use the small flat site in this case
