I have been watching some videos on how batteries work and they seem to talk about the cathode being the most important. I also watched another video a long time ago and while my memory is blurred it talked about impurities in the cathode being the only reason the electrical charge doesn't flow constantly. While I found this weird I also wonder how generates seem to get this unlimited supply of electrons also so it's only fair for me to assume that the battery can also.
I don't think that's accurate. Even the universe doesn't have an "unlimited supply of electrons".
Impurities in the cathode likely give rise to internal resistance inside the battery (among other things), leading to deviation from an ideal cell. I would guess that as your cathode approached purity, your battery would approach the behavior of 'ideal cells', and that discharge rates would approach those of capacitors, and efficiency would approach 100%. To get an ideal cell though, the anode and electrolyte would also have to be 'pure' and some real-world effects would have to be ignored.
Because of internal resistance, the terminal voltage of a cell that is discharging is smaller in magnitude than the open-circuit voltage and the terminal voltage of a cell that is charging exceeds the open-circuit voltage. An ideal cell has negligible internal resistance, so it would maintain a constant terminal voltage of $ \epsilon $ until exhausted, then dropping to zero. If such a cell maintained 1.5 volts and stored a charge of one coulomb then on complete discharge it would perform 1.5 joule of work. In actual cells, the internal resistance increases under discharge
What do you mean by pure, exactly? Complete homogeneity and lack of any other elements? That sounds like a physics problem where you "ignore air resistence" and other real world effects for simplicity. If this is what you're after, search for things like 'Ideal Battery' and 'Ideal Electrochemical Cell'.