Yes, electrolyte is an essential component because, as for dopants in semiconductors, it provides a source of electrons that support current flow. Without an electrolyte, wood acts as an electrical insulator and nothing happens: certainly no current flow. Anions carry a negative charge in the form of one or more surplus electrons in their conduction band, which are available to move under the influence of an applied emf, bumping each other along towards the positive electrode and allowing other electrons to enter from the negative electrode, and with an equal number of electrons being consumed by the positive electrode. Hence we have an electric current through an electrically neutral ionic solution. Why are you so blinkered?
With your ionic charge transportation system you have provided no mechanism for the anionic charge to be transferred to the positive electrode because there is none, nor any means for electrons to enter from the electrode: all done with smoke from where the sun doesn't shine and ionic mirrors I suspect.