Why must a voltaic cell take place in a conductive solution?
The electrolyte must be conductive to ions, not electrons. In fact, if there's an easy path for electrons to move between the two electrodes, the cell will immediately self-discharge. The exact reason for needing ionic-conductivity kind of depends on the cell. For example, in lithium-ion batteries, lithium-ions are generated at the anode and consumed at the cathode (or vice-versa). If ions cannot move freely between the two (through the electrolyte), then you will quickly build up an ion-rich region at one electrode and an ion-deficient region at the other at which point the reaction will stop.

It's not even a requirement that the electrolyte be liquid, as long as it conducts ions. Eg. the sodium-sulfur battery uses a form of solid alumina that conducts sodium ions.