When dimensions become very small, we are no longer in the realm of classical physics where potentials have mathematical singularities as in the classical 1/r Coulomb potential. Point particles belong to the realm of quantum mechanics and there the laws and computational rules are different.
The electron is a charged point particle in the standard model of particle physics, but when interacting with a proton, the potential near the point location is not accessible. What happens the electron is bound to the proton forming the hydrogen atom, and it cannot fall on the proton as the classical 1/r potential indicates. There is a lowest energy state, and it stays there in orbitals, which are instead of a track, probable locations where it may be found.
So it has no meaning for quantum dimensions to be asking the value of the 1/r potential. Its functional form has been used to calculate the orbitals, and even when there is a probability of overlap, as seen in the S orbitals, there are quantum number conservation rules and energy conservation rules that keep the orbitals stable.
Actually, the stability of the atom is one of the reasons that quantum mechanics had to be invented. In classical physics because of the 1/r attraction the electrons would fall on the nucleus, neutralizing them, and no chemistry would exist.
The proton cannot become neutral when the electron overlaps with it in space, because quantum mechanically the neutral particle is the neutron, which is heavier , and also electron lepton number has to be conserved. ( there is electron capture in nuclei where there is extra energy to allow for the reaction , but that is another story)
One needs to study quantum mechanics and nuclear/particle physics at the realm of point particles.