Chosen Solution

This mac simply stopped dead, no previous symptoms whatsoever. Diagnostic LEDs indicated faulty logic board (#1 lit, #2/#3/#4 off). Replaced logic board with refurbished one, but with no effect (LEDs same as previously). So my question is - could the problem be the power supply? In others words, could LED#1 light even if there was insufficient power getting to the logic board? If this is the case, would it be worthwhile purchasing a used working power supply? (the refurb logic board wasn’t too expensive, but I’m reluctant to spend much more on an old computer if there was no certainty I could get it working) Update yes, I did enable the PSU with the jumper pins as indicated in the article you mention.

From the G5 17" (non iSight) manual • LED 1 indicates that trickle voltage from the power supply has been detected. This LED will be ON when the computer is turned off and your power supply is working correctly. • LED 2 indicates that the main logic board has detected proper power from the power supply when the computer is turned on. This LED will be ON when the computer is turned on and the power supply is working correctly. • LED 3 indicates that the computer and the LCD display are communicating. This LED will be ON when the computer is turned on and video signal is being generated. • LED 4 lights only if the computer detects an over-temperature condition. This LED will be OFF when the computer is turned on and running at the correct temperature. Assuming I am reading the manual correctly, If LED1 is ON then the PSU is at least powering up to some degree. Check THIS PINOUT guide or THIS ARTICLE and see if the other voltages are present. Note the instructions to power ON the PSU while not connected. THIS ARTICLE may also be of interest

If just #1 lights up, this indicates a faulty power supply. I have no idea why you replaced the logic board. This model had a lot of issues with bad capacitors so I would inspect them for corrosion and or swelling.