Completely not turning on at all? Then there's something pretty bad going on with the card.

Normally they don't get detected and the motherboard complains.

Correct, when the card is physically installed into the board, the PC no longer even powers on let alone POSTs.

One (or possible mode) of the power connections is probable shorted to ground - and the PSU self-test/overload protection is triggering which would it into power-down mode. (Often when this happens. you can briefly see LEDs flash, fans jiggle etc. - just signs that on some rails power came up briefly).

With the card removed, I'd measure the power pins to see if there is a direct short to ground (keep in mind that being a "high power" these may have a fairly low natural impedance path to ground).

You're looking for something of such lower resistance that it would exceed the PSUs rating for that rail. Another possible indicator is low-resistance to ground at both meter polarities (high power semiconductors typically only have a unidirectional natural conduction state - just use a good meter which doesn't have high voltage/current for resistance measurements.

If you can detect such a thing (and even it not - just knowing which rail might give a better indication of area to look) - carefully examine the card components up-close and look for charring, discoloration or any other indicators of excessive power draw by that component.

So, in a twist of events, the card no longer causes motherboards to fail to power on, and there doesn’t appear to be any shorted power rails either on the AGP pins or the PEG connection. The only “change” at this point was that I removed the backplate and heatsink to get a good close look at the card, I could see no obvious damage nor missing SMD components, so I rebuilt the card.

The behaviour now is that with the card inserted and the PEG power connected, the system will power on, but not POST nor show anything on the display. The keyboard lights flash once, but there are no signs of life. I have a POST analyser card, but this motherboard (Asus A7N8X deluxe v2.0) doesn’t appear to send codes over PCI, even when using a working card I don’t see POST codes.

If I disconnect the PEG power connector then the motherboard POSTs. I get the expected single beep, and numlock/caps lock/scroll lock function, as does graceful power-off by single pressing the power button. There is still nothing on the display however, when I’d have expected a power connection error message.

It’s certainly looking like it’s unrepairable.