Electrons' flow misconception?
Batteries store chemical energy.  The two different metals and the electrolyte react in such a way that electrons will flow through an external circuit from the negative terminal of the battery to the positive terminal.  As the reaction proceeds, the metals are converted to another material and the electrolyte becomes weaker until finally, the reaction stops.

For example, in a charged lead acid cell, we have:



The electrolyte reacts with the lead plate to convert some of it to lead sulfate along with some excess electrons.

The electrolyte reacts with the lead oxide plate to convert some of it to lead sulfate along with a deficit of electrons.

With an open circuit, the reactions are stopped due to the charge accumulated on the plates.

When an external circuit is present, the excess electrons from the lead plate flow through the external circuit to the lead oxide plate but this allows the reactions to proceed anew and the process continues until the cell is discharged:

  

So, the battery drains because chemical reactions drive the electrons through the external circuit until the reactants are exhausted.

(images from Wikipedia article for "Lead-acid battery")