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?