In this paper on Page 21, the first line of the proof, Peter Scholze seems to claim that any hypercover, consisting of finite sets, splits. I find this hard to believe.

I am not familiar with categorical topology, but let's consider the constant simplicial set, where you map all simplices to a point and all arrows to the identity. It seems to me that this is a counterexample, is it not? And if not, why is Scholze's claim true?

EDIT: I need to explain my question a bit: According to this source, a simplicial object $U_\bullet$ splits if for every $n$ there exists a subobject $NU_n$ of $U_n$ such that the map $$ \coprod_{\phi:[n]\twoheadrightarrow [m]}NU_m\to U_n $$ is an isomorphism. In my example we have $U_n=*$ the one pointed set. It has exactly two subobjects, the empty set or the point. The subobject cannot be the empty set, because then the coproduct (=disjoint union) would be empty, too. So it has to be $*$. In wich case we get an isomorphism between the coproduct of several copies of $*$ and $*$, which is absurd.