The problem you are facing is based on an idealization of wires as perfect conductors. $\Delta V=0$ is true only if the resistivity of the wires is zero. You have two ways to resolve this apparent paradox: either you consider a limit approach to infinite conductivity, where the electric field in the conductor $E=j/\sigma$ approaches zero in the limit, or you consider the small but finite conductivity of real wires. In this latter case the field E in the wires is not zero. It's just very small and the voltage drop across any portion of wire is negligible.