Reference for things "locally looking like" objects in a site?

Question

Given a site $\mathcal{C}$ (with possibly extra structure), is there a good notion of a "completion" $\overline{\mathcal{C}}$ of $\mathcal{C}$ which results in a site of things that "locally look like" objects of $\mathcal{C}$? Specifically, in nice situations it would be ideal if anything resembling the following (especially the equivalence between 4+5 and 6) hold:

1. $\overline{\mathcal{C}}$ is a site,
2. $\mathcal{C}\subseteq\overline{\mathcal{C}}$ is a full subcategory with no more covers than $\mathcal{C}$ originally had,
3. $\overline{\overline{\mathcal{C}}}=\overline{\mathcal{C}}$,
4. An object of $\mathcal{C}$ can be given by a specification of Čech/torsor datum, i.e. collections of objects $\left\{X_i\right\}$, "open" subobjects $\left\{U_{ijk}\subseteq X_i\right\}$ such that the allowed indices $k$ depend only on the unordered pair $\{i,j\}$, and isomorphisms $\varphi_{ijk}:U_{ijk}\rightarrow U_{jik}$ satisfying a cocycle condition,
5. Every object $X$ has at least one specification in terms of this sort of datum, and for any such specification the objects $\{X_i\}$ cover $X$ in $\overline{\mathcal{C}}$,
6. An object of $\mathcal{C}$ is also the same as a sheaf on $\mathcal{C}$ admitting a "cover" by representable sheaves, and this is a covering in $\overline{\mathcal{C}}$,
7. $\operatorname{Sh}\left(\overline{\mathcal{C}}\right)=\operatorname{Sh}(\mathcal{C})$,
8. If $X$ is a scheme and $\mathcal{C}$ is the Zariski site of (opposite) quasicoherent $\mathcal{O}_X$-algebras, then $\overline{\mathcal{C}}=\mathbf{Sch}/X$,
9. If $k\in\mathbb{N}\cup\{\infty\}$, $\mathcal{C}$ is the site with one object $\mathbb{R}^n$, $C^k$ maps as morphisms, and topological open coverings by injective local diffeomorphisms as covers (or some site of this sort), then $\overline{\mathcal{C}}$ should be $n$-dimensional $C^k$-manifolds,
10. If $\mathcal{C}$ has the indiscrete topology, then $\overline{\mathcal{C}}=\operatorname{Sh}\mathcal{C}=\operatorname{PSh}\mathcal{C}$ via the Yoneda embedding. Thus, for example, if $\Delta$ is the simplicial category then $\overline\Delta$ is the category of simplicial sets.

I can find many references making the functor formalism 6 for this precise in special cases of #8 that should easily generalize to all of #8, but I can find no reference which tries to make the construction of such completions systematic, or abstractly describe the equivalence with the Čech viewpoint. (In the scheme case, the equivalence between 4+5 and 6 basically consists of a standard representability lemma for schemes, e.g. Stacks 26.15.4.) Criterion 7 also makes me wonder if this can be thought of as some adjoint to the functor $\operatorname{Sh}$ from sites to (Grothendieck) topoi. For reference, I know essentially nothing about topoi.

For convenience, I am pretending there is no such thing as a "set-theoretic issue."