A few recommendations on this. I tried to reactivate 2 of my AirPort Extreme A1408 today. One is still working althought nearly 10 years old, the other comes up with an amber static light. Power supply is not the problem as I tried the 2 power supplies crossover. There are a few points I could think of what the problem is: (1) Faulty capacitors. A problem that reoccures every few years just because some bean counters buy cheap parts. (2) What I think is much more likely is the fact that the battery is dead. In my case a reset was working and now I have a base station that does not store any settings. And indeed storing the settings is supported by the battery. I have seen this on a lot of very old Macs that run quite well but at startup they tell you that time and date is not set correctly. Measuring the batteries have always shown that they were empty. Maybe this helps. I am not quite sure if I should repair or try to repail this old device. If I do I will send additional comments here. Update...

It looks like your iMac has the same problem as one a friend bought. If the capacitors show no damage than the most possible problem is the NVidia graphics chip. There were a few with bad soldering balls. This means that the iMac disconnects the graphics bus during running due to bad soldering. Some run 1min, some 5min, some 15min. I currently try to fix this problem by heating up carefully the area below the NVidia chip. Other people were sucessfull by putting the whole motherboard inside an oven at 250°C for 10min covering all of the board excluding the area with the NVidia chip with aluminium foile. At ebay Germany there is a company doing professional repair with a 20k$ ERSA soldering robot. But this is approx. 300$, a price you can get a working iMac 20". Apple had a exchange program but it ended a few years ago as far as I know. You can search for baking nvidia cards and you find a lot of solutions. Good luck.