Does a higher concentration of electrons mean a stronger electric field in a battery?

Yes.

Electric field strength and potential difference are closely related, when the electric field arises from excesses and/or deficits of charge. It works like this.

To say there is an electric field between the terminals means that if you put a charge at some point (e.g. inside a wire or resistor connected between the terminals) then there is a force on that charge. It is this force that urges free electrons through the wire and causes an electric current.

The potential difference between the terminals is the potential energy lost per unit charge by a (positive) charge going from one terminal to the other. That is to say it equals the work done on it by the electric field as it goes from one terminal to the other.

But work = force $times$ distance (in direction of the force), so the greater the electric field strength, the greater the potential difference.

Yes. Note that the electrostatic force is very, very strong. So you only need a very slight decrease in spacing between electrons (and a corresponding increase in concentration or density, if you will) to get an enormous increase in the repulsive force and corresponding electric field.

See this Srendi and Matt's answer in this question since they understand it a lot better than I do: https://electronics.stackexchange.com/questions/515653/what-actually-happens-to-electrons-in-a-circuit-when-work-is-done-at-a-component/515733#515733

In the Fenyman lectures which you can read here: https://www.feynmanlectures.caltech.edu/

It claims that if the balance of electrons to protons in your body was off by 1%, then the force produced would be enough to lift the Earth.

So the change in concentration/density is really, really tiny.