In a voltaic pile with copper and zinc as the two metals and with aqueous sodium chloride (salt water) as the electrolyte, what happens to the electrons and ions and what reactions occur?

I understand that in a copper/zinc voltaic pile using sulfuric acid as the electrolyte, the zinc electrode decomposes into positive zinc ions (which dissolve into the electrolyte) and 2 electrons, which travel through the exterior wire to the copper electrode. At the copper electrode, the electrons are then accepted by positive hydrogen ions from the electrolyte solution which form hydrogen gas at the copper electrode.

The problem I see when this is adapted to a sodium chloride electrolyte is that the positive deposited ion would be pure sodium and it seems unlikely that sodium would form in its pure form both because zinc cannot reduce sodium (being lower in the reduction potential list) and because sodium is very hard to obtain in its pure form. Nonetheless, using a sodium chloride electrolyte for a voltaic pile seems to work frequently in many household experiments (according to my research),

So my question is: What is going on? Is there a new, different reaction? What happens to the ions and electrons?