That twitch-and-die syndrome is almost always due to bad cap(s) in the power supply. The stand-by trickle current is enough for the power switch to activate the start-up circuit, but the main switchmode circuitry of the PSU is not kicking in. The most likely culprit will be a high voltage cap in the circuit before the main switching transistor(s), but if you are going to the trouble of changing one, you might has well change the lot.

The eMacs that are dying in large numbers are most likely the ones that were left switched on 24/7 in school computer labs and suchlike - the capacitors seem to have a certain life in hours x heat stress. I have a couple of older eMacs, ex office and home use, that are running perfectly. It's a bit of an ugly duckling but the display's are still pretty good by any standard.

Rick