The collision rate probability decreases when using compression?
No. On the contrary, if we model $H$ as a random function chosen independently of $Z$ and $S$, then $P_c[H\circ Z](S)=P_c[H](S)$.

Argument is that $P_c[H](S)$ is dependent only on the width of $H$ and the number of elements in $S$, as long as $H$ is modeled as a random function chosen independently of $S$. Define $S'$ as the set of $Z(s)$ for $s$ in $S$. It holds that $P_c[H\circ Z](S)=P_c[H](S')$, and because compression is reversible, $S'$ and $S$ have the same number of elements.