I specifically used a scenario in which the charge did NOT increase on the plates. In fact, you could increase the charge on the plates and still see the voltage go down. And you can have a material in between the plates that starts with a low dielectric constant and then, through a chemical reaction, ends up with a high dielectric constant such that, without doing anything to the system, the charge on the plates remains the same but the voltage goes down.As I mentioned above: "other things equal". If you take a capacitor with dielectric or without it, and increase charges on its plates, the voltage will go up. You do not insert dielectric into a battery, you do not change distance between its electrodes, so these things remain equal for the battery. In these conditions only charge separation will dictate the strength of electric field of the battery, and hence the potential difference, that is the voltage.