If that PSU is as old as the CPU (10 years?) it's probably on its last legs. I would replace that before it fails and takes other parts with it.
If sure about system not over heating and If available I would check with another working PSU and see if shutdown still happens or not.
PSUs lose some of their nominal capacity over the years (some percent each year) and the quality of design, buid and components (SMDs etc) has a direct correlation with their degradation. As said above, that's not a great PSU to begin with.
Not sure about the 550W CM MWE (didn't see any credible review if there are) but it's better than the current one I guess.