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?
Without a ground there is nowhere for a surge to go, permanent damage is much more likely. Surge protectors or a UPS will not protect against surges at all without a ground.
There’s also no ground so the chassis may have enough voltage on it to cause a shock if you touch it. This could also damage components as they are not grounded and touching things can introduce high voltage from static electricity which will have nowhere to go.
Additionally if you have ethernet connected to it the system may end up grounding itself through the ethernet cable, if the device at the other side does have a ground, which could cause issues.
So it basically just means you have a much higher chance of damaging the parts, or injuring someone touching things.