I would argue that correct intuition about anything can be had without math. Using math may be one way of gaining that intuition, but intuition, by its very nature, is not mathematical.

I'm not sure it's useful to argue whether an intuitive picture of electrons hopping or not is more correct. (Electrons are in any case not really little balls that are in one place.) These are all models of the world made with specific goals, where the most "correct" model would be solving the schrodinger equation for a macroscopic system -- something that would not give you any particular insight as to how things behave.

I think the usefulness of an intuitive layman's understanding should be derived from whether it makes you conclude the correct things about the macroscopic behavior of electricity.