In metals, such as copper, electrons are free to move. They are mobile. They stay attached to the wire, but they can move around within the wire. What causes DC current to flow is that a voltage potential is applied across the wire. Because of how the universe works, the voltage across the ends of the wire creates a uniform electric field (voltage gradient) along the length of the wire. The electrons, since they are charged particles and free to move, travel along the wire "feeling" the electric potential gradient. But the number of electrons in the wire remains constant. For every electron departing the wire at one end, another one is being inserted at the other end. That is how circuits work at a basic level. An electric field causes charges to move in a conductor.
If there is a field, but one end of the wire is open-circuit, so no current can flow, then the electrons kind of crowd one end of the wire or the other as you say. But that doesn't really happen when there is a closed circuit. Or doesn't happen to any extent that you need to worry about for practical purposes when there is a closed circuit.
I suppose the superconductor is an exception because charges can move in a super conductor even without an electric field.