Dell OptiPlex 755 - sometimes after plugging in the power, power button lights solid amber, fans spin at full speed and the machine is dead. When the power is removed and plugged in again (sometimes), button becomes green and the machine works fine.
The power supply is good (working fine in another OptiPlex 755).
With another known good power supply (from another OptiPlex 755), the problem becomes permanent (it does not boot no matter how many times the power is cycled).

The failure is on the motherboard, probably something with POWER_GOOD. This signal goes to SMSC SCH5514E-NS, but there's no datasheet or schematic available.

Is the CMOS battery low? Have you tried resetting and reflashing the BIOS? Since the machine is capable of running normally, a real hardware problem is not very likely. Chances are it's some kind of corruption in the firmware.

The battery is new (and OK). We have hundreds of these machines and they start even with bad battery. Some of the batteries are discharged after 13 years, mostly where the machine was not used for a couple of years.

The CPU probably does not even start executing BIOS code because some reset signal is in an incorrect state. Schematic or SCH5514E-NS datasheet would probably help.

When you first plug it in, it turns on and runs for a couple of seconds: that's when the BIOS uploads the firmware into the embedded controller. So an issue with BIOS could potentially disrupt the power sequence in some weird way that would be a nightmare to debug. It makes no sense to do anything else until you flash in a known-good BIOS. Make a backup of the old one of course.

There are no datasheets for these Dell ECs that I know of but there are a few public schematics featuring the chip. The closest match is probably the Optiplex 360.