The energy transported by current is not stored in the motion of the electrons but in the field they create. So you can make them go back and forth or slowly drift in one direction. The source will make them move in the wire near the source this starts to creates a field, the field propagates along the wire at close to light speed and gets them moving everywhere. If you plug something in the circuit this disturbs how the electrons can move near it. This causes the field to be different. The source then has to do work to keep the field stable. This is how the energy from the source gets whatever you want to power.
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