No need for new cat, just get a muzzle for the current puss. ha, ha.
It is correct that many components share a common ground via the interconnect grounds, but not all circuits are also referenced to the chassis ground. In other words the IC's may create a ground only between the circuit boards while the boards may be floating and not referenced to the chassis. The externally added ground wire will make sure the two chassis have the same ground reference.
A good way to tell if the RCA jacks and corresponding circuit are referenced to chassis ground or isolated is to look closely at the RCA jacks. If you can see the metal of the jack making a direct metal-to-metal contact with the chassis and doesn't have any insulated washers or shoulder washers around it, then you have a chassis grounded jack.
Now, even though the RCA jack is grounded directly to the chassis originally, there is no guarantee that the ground aspect of the connection is still good after time. Aluminum corrodes, steel rusts, copper oxidizes, and powder coatings can interfere with a good electrical ground contact. Look for star type washers that have teeth that penetrate the chassis metal for the best and longest lasting connection in this situation.