No doubt the watch battery on the motherboard long since lost its juice.

However, I'm not quite sure what error I saw on boot, I think it was before POST and I think it mentioned "low voltage". A quick search of dmesg shows nothing with regards to "voltage", so it was probably before POST.

I'll do what's feasible with regards to looking at the power supply, but any suggestions? It's a "mature" PC (meaning quite ancient with a retail value of zero or fewer dollars).