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?
The most important risk you face is if somehow mains voltage ends up contacting somewhere you get electrocuted and die.
There are 2 purposes of an earth ground: First it can be used as a reference for certain signals, such as microphones. Second, it can be used to protect against turning yourself into a sparker.
There is a clear separation between mains voltage and system voltages so it’s typically not going to be a problem, but if a little wire ends up contacting the power supply case it can become energized and things start to get really bad.
Most of the electrical code where I live focuses on grounding as “Bonding”, which is purely safety related for giving dangerous voltages a safe place to go.