Understanding current through batteries and Kirchoff's Law
Current flows through batteries, electrons do not...

Batteries are sort of like bottomless capacitors, continually refilled with charge by a chemical reaction.

When in circuit, electrons leave the negative terminal and are immediately replaced by an electron entering the positive terminal. The arrival of this electron causes the chemical reaction to advance which in turn creates a new electron ready to leave at the negative electrode.

The whole process repeats until all the chemicals involved in the chemical reaction are expended and the battery is dead.

However, while still alive, since the number of electrons leaving the negative terminal equals the number arriving at the positive terminal, we can say current flows through the battery.