I think that right there is your issue. If I were to build a PC with that calculation, I would have, at minimum, gone for a 800W PSU. Mainly for provide a safely margin and also allow for some expansion down the road.

A PSU that is that old isn't likely to be able to keep its peak wattage like it used to. The old electrical components age. I would guess, whatever protection circuitry kicks off the power supply if the power draw goes too high, has likely aged enough to lower that kick-off point. While it could handle 650W while new, maybe now it shuts down at something like 620W.

Your only real option may be to just get a new PSU. As a temporary measure, you could maybe limit the power draw of the GPU. I don't really know how to go about it, but the TDP limit on the Steam Deck immediately came to mind.

If that's true, you definitely need a new PSU. You don't want to run that close to the upper limit of power. PSUs, like everything, degrade over time. A 10 year old PSU almost certainly can't maintain max power reliably.

If you want something that'll last and allow room for expansion, I'd get a 1000W PSU.

For estimates you can subtract about 10% of max wattage per year the PSU has been running, which is a pessimistic take but possible depending on quality of the PSU. I am astounded you can run a stress test without crashing.