From a naive student standpoint, something is seriously wrong. I have no answer to this. As far as I know, it is a brand new physical phenomenon. And besides, if you charge up a bowl of small metal parts, then pour the charged parts out in a stream, you definitely have e-fields and surface charge, and you definitely have an electric current as well. The same is true of electron beams in vacuum, charged paint from electrostatic sprayers, charged rain in thunderstorms, belts on VandeGraafs, etc., etc.
Electric current appears in the connecting wires.
Now break the circuit. The flowing charges do not zoom back into the battery, instead they halt where they are. There is no be problem in accepting that an imbalance of something might be able to move along. If the attracted lint and crackling sparks are caused by imbalanced charge, then we won't become confused when charges start flowing and stop. Maybe my ideas can be clarified by an analogy with Newtonian Mechanics.