One of the coolest things about superconductors is "kinetic inductance"[1]. There is no energy in a normal current. As you say, the electrons don't travel all that fast most of the time. They feel a field, accelerate until they bump into something and start all over again. That's resistance; the energy stored by an inductor or capacitor (a current or voltage) is in the magnetic or electric field nearby.
This isn't true for Cooper-pairs in superconductors. They go fast enough that there is true energy stored in their velocity; there's nothing to stop them, they're a superfluid. This is used for certain types of detectors (MKIDs), and means that there is a bit of extra L in superconducting transmission lines, above the geometric inductance.