Um, assuming there are no ground loops, and perhaps no other ground path, then the shield would absolutely become part of the circuit. Consider devices without a ground pin, otherwise yes, current would be distributed exactly as stated above.
If I’m paying for pure silver conductors, or fancy litz, etc. with an aluminum foil shield and AL drain, then I would want to avoid having any current in that shield at all. To do that I must avoid connecting the drain at both ends.
Now, what you may like to listen to is another story. :)
Best,
Erik