Over time electrolytic capacitors in PSU can degrade and have negative effects on PSU's power output. It could happen that they are in so degraded condition that PSU can't provide enough power, even if it could do so at one time in the past.
As for the shutdown, it could be cased by several factors. PSU could have protection circuitry which could detect if it's overloaded. Motherboard could also be causing the shutdown. Basically, when the computer is started, it takes some time (usually in millisecond range) for PSU to stabilize. There are voltage sensors which on the motherboard which will prevent computer from working if the voltages aren't good enough for it to work safely. It could be that PSU is tripping that sensors which could be shutting down the computer.
It could also be overheating issue with the PSU or some other component. If you haven't done so, clean the insides of the PSU with compressed air or something similar. There cold be tons of dust inside which can be causing overheating.