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Basically, charge and energy are not the same thing. Both are flowing when there's a complete electric circuit, but while the charge moves with the current. along the wire, the energy flows perpendicular to the charge, into the electromagnetic field.
The complete circuit means electrons begin to flow. Moving charge creates a magnetic field. Energy flows in this magnetic field. Energy is not flowing along the wire. It's flowing radially out of the battery, perpendicular to the wire, and into the device. The energy flows through space. The current merely creates the electromagnetic conditions needed for the flow of energy to be able to happen through space. The device takes energy out of the EM field, and the battery replenishes it. When the battery has no power left, electrons stop moving, and the EM field collapses. Energy flow ceases.
It's all a bit crazy though, not remotely analogous to water flow.
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