The disagreement may have come from mutual misunderstanding.
Unless a cell has lost some matter, it contains the stable count of electrons, safe random fluctuations and eventual net charge caused by external electrostatic potential.
Aside of that, some of these electrons relate/belong to electro-active substances taking part in electrode half-reactions, ...
Your professor is wrong, and you can prove it.
According to your "professor":
As a battery is used, through the flow of electrons, electrons are lost to the environment ... Those electrons are no longer in the battery. Thus, the battery has the same number of protons and neutrons, but less electrons.
which implies the charge of the battery would ...
Very bad explanation in the email response.
The explanation reads..
"Thus, the battery has the same number of protons and neutrons, but
less electrons. This also means more unreactive metal cations exist in
a used battery."
No, not at all. An electrochemical cell is not ionized. It is always neutral overall. However, the electrodes are indeed ...