Current flow in batteries?
The confusion here is from the initial poor description of how a battery works.

A battery consists of three things: a positive electrode, a negative electrode, and an electrolyte in between. The electrodes are made of materials that strongly want to react with each other; they are kept apart by the electrolyte.

The electrolyte acts like a filter that blocks the flow of electrons, but allows ions (positively charged atoms from the electrodes) to pass through. If the battery is not connected to anything, the chemical force is pulling on the ions, trying to draw them across the electrolyte to complete the reaction, but this is balanced by the electrostatic force-- the voltage between the electrodes. Remember-- a voltage between two points means there is an electric field between those points which pushes charged particles in one direction.

When you add a wire between the ends of the batteries, electrons can pass through the wire, driven by the voltage. This reduces the electrostatic force, so ions can pass through the electrolyte. As the battery is discharged, ions move from one electrode to the other, and the chemical reaction proceeds until one of the electrodes is used up.

Thinking about two batteries next to each other, linked by one wire-- there is no voltage between the two batteries, so there is no force to drive electrons. In each battery, the electrostatic force balances the chemical force, and the battery stays at steady state.

(I kind of glossed over what it means for two materials to "want" to react with each other.  Google "Gibbs free energy" for more details on that. You might also google "Nernst equation.")