Why does the cathode have to be made out Cu in a Zn-Cu voltaic cell?
You could certain have a zinc-copper cell in which copper was not the cathode material.  Graphite or silver or whatever could be the electrode material instead.   But once you close the circuit:


Current flows
Cu is deposited as a metal solid on the electrode surface, whatever it is made out of.
Copper metal is, as it always is, conductive.
As a result of steps 2 and 3, there is now a copper metal cathode instead of a cathode made of another material.


So even if you start without a copper cathode you wind up with one.