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?
You are correct that this is technically in code and would protect against shock hazards in a neutral error situation but you also get the opportunity for the outlet to pop during the day when nobody is home and the battery to die.
We had a situation in our old house where someone who was technically correct but didn’t think it through had a gfci outlet upstream of the refrigerator outlet. Thankfully it popped while someone was home and we got everything corrected before we lost everything in the fridge.
For sure, a fridge is a really bad one to be using on an actual ungrounded GFCI, exactly for the reason of risk of expired food.
I like my homeserver but if something trips and its offline for a while it’s not gonna ruin my day.
You food should expire that quickly. It would take at least a day or two.
Unless your gone for a while you would notice that the lights no longer on and it doesn’t feel as cold.