I know that the answer is yes, I should, but outlets near the setup are not grounded (even though they look like they are) and I don’t want to have wires running though my living room.
The real question is what are potential problems ? Occasional system reboots? Permanent damage to PSU? Permanent damage to other components?
Run the ground into the ground. You can get metal wire that would work nicely.
Except in very rare configurations (i.e. not 99.9% of residential), you do not want to have multiple paths to ground within a system. All grounds should go to the tied ground/neutral bus in the main breaker panel, which then goes to earth via a ground rod or a clamp to a copper gas/water line, etc. Otherwise you can have current flowing in ways that the system isn’t designed for, which at the least can trip breakers and GFCIs, and at worst exceed the rating of the wires in a short condition and cause a fire.