"Which is more important to how a Zapper works, the voltage impressed on the body or the electron flow through it?"
good question. the amount of current actually involved in killing pathogens is probably very minute, i.e. micro-amps or less rather than milli-amps. the trouble is it's non-zero and must be evenly spread to the whole body. since induction/impedence is higher per amount of current in an oscillating field, you can get away with far lower current on a pulsed signal than straight dc.