I think my basic concern is this: to build up a charge separation inside the battery (that ultimately is responsible for the potential voltage difference between electrodes........
A charge separation is not responsible for the potential difference. The potential difference is inherent in the chemical entities that are interacting. As best as I can explain it the charges are transferred because one element of the chemical process has a higher attraction for electrons than the other. The valence electrons of one element is more loosely bound than the other. Take the simple Daniell cell that uses Zn and Cu as the interacting elements. On discharge Elemental Zn more easily give up it valence electrons to Cu ions resulting in a lower overall energy of the pair. If you put metallic Zn in contact with Cu ions as in a CuSO4 solution the Zn will dissolve loosing its valence electrons to the Cu ions. In the Daniell cell the Zn is isolated from the Cu ions. When the Zn cathode is connected to the Cu anode the affinity of Cu for the Zn electrons is transferred via the conductor because of the potential difference inherent in the pair.