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.
Horizontally Rotating Rat