Maximum current is drawn from a cell when it is shorted. But, why?
Here is the false assumption:


  So, for a given cell, I assumed V to be constant.


In a real battery, the voltage decreases as current is drawn. A real battery is complicated, but we often model a battery as an ideal voltage source with a series resistor (called source resistance). That is the small r in your circuit above.

The ideal source voltage (E) is constant, but the actual voltage that you could measure across the battery terminals is not.

By the way, it's worth looking at a battery datasheet, such as Energizer E91 AA Alkaline Battery. You can see from the graphs that the real battery behavior is nonlinear, but those curves can be approximated as a line with negative slope -- ideal voltage under no current, and less voltage as current is drawn.