As predicted by Maxime, the argument in Nikolaus and Scholze is indeed incorrect, but it can be fixed as follows. Note that since each map $(1/k)\mathbb Z \to (1/n)\mathbb Z$ factors through some $\mathcal C_{[s,s+1)}$, we have that for $b>a+1$ a pushout of simplicity sets $C_{[a,b)} = C_{[a,b-1/n)} \cup_{C_{[b-1,b-1/n)}} C_{[b-1,b)}$, which allows to show by induction on $b-a$ that $C_{[a,b)}$ is contractible. Then we can exhaust $C$ by intervals of increasing size to see that $C$ is contractible too.

As predicted by Maxime, the argument in Nikolaus and Scholze is indeed incorrect, but it can be fixed as follows. Note that since each map $(1/k)\mathbb Z \to (1/n)\mathbb Z$ factors through some $\mathcal C_{[s,s+1)}$, we have that for $b>a+1$ a pushout of simplicial sets $C_{[a,b)} = C_{[a,b-1/n)} \cup_{C_{[b-1,b-1/n)}} C_{[b-1,b)}$, which allows to show by induction on $b-a$ that $C_{[a,b)}$ is contractible. Then we can exhaust $C$ by intervals of increasing size to see that $C$ is contractible too.