Here's a simple explanation: https://depts.washington.edu/matseed/batteries/MSE/components.html
Basically electrons flow from the anode to the cathode and that discharges the battery. Recharging is the process of herding the electrons back through the electrolyte to the anode.
And we want to do this:
a) quickly, so we can charge in minutes, not hours
b) efficiently, without "using up" any of the materials involved
c) without releasing too much heat either
That's a tall order, if there ever was one. So yes, chemicals and chemistry as we know them today most likely aren't the solution we're looking for.