Snot random at all. Its faradays law, but for alternating current.
"EMF in a closed loop is proportional to rate of change of magnetic flux', or something, iirc. The change in flux is caused by the motion of a ferromagnetic object within the field lines (the string), and as the flux changes it induces a current in the coils. The coils, simply sitting there doing nothing, do not change the magnetic field and the change in the magnetic field is not caused in this case by the current in the coils, its caused by the motion in the string. The same reason that the current in the coils cant increase the fields strength (the current us generated by the changes in the field).
In the case where your driving current through a wire thats perpendicular to a magnetic field, the field will increase in strength. Thats not happening in a pickup. They arent electromagnets.
Conservation of energy is the main...no, only reason for not having infinite sources of energy. Not efficiency.