Consider a simple circuit with a battery and a resistor, connected by wires.
We usually treat the wires (excluding the resistor) as ideal conductors (no resistance). Therefore, we conclude that the voltage drop happens across the resistor only, meaning that the potential across any two points of the same wire (either from one end of the battery up to the start of the resistor, or from the end of the resistor to the other end of the battery) is the same.
If the potential across a wire is constant, and the electric field is the derivative of the potential, it means the electric field on any point on the wire is zero.
If the electric field is zero, it means no electric force is acting on charges in the wires and accelerate them.
The charges in the wires do move, though, so it appears electric field does act on them. What am I missing?