The topology you mention is called the "closed topology" $J_\mathrm{cl}$ in "Points in algebraic geometry" by Gabber–Kelly. See also the comparison diagram by Pieter Belmans.
Note that Gabber and Kelly additionally ask that the covering families are finite. But since they ask for the base scheme to be Noetherian (and separable), I don't think there is a difference, because any covering family automatically has a finite refinement.

You can apply the Comparison Lemma to simplify the site $(\mathrm{Sch}/S,J_\mathrm{cl})$ without changing the category of sheaves on it. To make the notation easier, I will consider $J_\mathrm{cl}$ on $\mathrm{Sch}$, but the argument can be modified to work for $\mathrm{Sch}/S$ as well. The claim is that the category of sheaves $\mathbf{Sh}(\mathrm{Sch},J_\mathrm{cl})$ is equivalent to the category of presheaves $\mathbf{PSh}(\mathrm{IntSch})$ on the category $\mathrm{IntSch}$ of integral schemes.

Any scheme $X$ can be written as a union
$$X = \bigcup_{x \in |X|} \overline{\{x\}}$$
and each of the closed subsets $\overline{\{x\}}$ can be given the reduced induced subscheme structure. In this way, any scheme can be covered by integral (i.e. irreducible and reduced) schemes. The Comparison Lemma then says that the category of sheaves for your topology can equivalently be described as a category of sheaves on the full subcategory of integral schemes. The Grothendieck topology that we have to take on this category of integral schemes is the restriction of the one on schemes: we again look at covers by closed subschemes. But for an integral scheme $X$, if you have such a covering by closed subschemes, one of the closed subschemes will contain the generic point, and then this closed subscheme must be equal to $X$. In this way, you can see that the restricted Grothendieck topology is precisely the presheaf topology.

In the same way, you can also find $\mathbf{Sh}(\mathrm{Sch}/S,J_\mathrm{cl})\simeq \mathbf{PSh}(\mathrm{IntSch}/S)$.

notgiven by closed immersions, but roughly speaking, by the formal completion along some closed subset (whose complement is quasicompact). $\endgroup$3more comments