Sounds like it's time to test the card in another pc.

Interesting. Thanks for that information.

Do you have a spare, or another pc, power supply you could try in your system instead of the evga power supply?
Can you try the video card in another system to see if it works?
Also, do you have another pci-e X16 slot to try it in (I know you got a ehole new motherboard, but just in case the evga power supply fried the main x16 slot. only install in this slot after you remove the evga power supply and put a different one in).
Have you tried using a different port? Display port, dvi port, vga port, hdmi port?
Have you disabled 'non-os related updates' in MS update? If not, please do right away.
Have you uninstalled ANYTHING to do with the video? Like gpu-z, cpu-z, ALL monitoring software, PrecisionX or MSI afterburner? Then use DDU to clean again in safe mode.
Then install an older driver, like 388.00 (the 365.xx drivers were the last before the 1000 series cards came out).
Also, can you use other cables that go to the video card? It sounds like you may have a bad cable from the power supply, or a bad power supply even though you monitor it.
 
In the bios, do you have the pci-e slot set to Auto, or Gen 1, or gen 2, or gen 3? Try those different options. Gen 3 should give you the most power to the video card. Also, is there an option for video power on bootup (I know my asus mobo has one, I forget the name though, but it can boot in low power mode or regular mode). Try theat setting (sorry I cant remember the name or location of it).
Also, go through the bios to see if there areb any other settings, like booting to pci-e device.
If none of that works, do the ddu uninstall of everything, unhook the power to the wall for both system and monitor, pull EVERYTHING except the cpu out of the motherboard with the power supply unhooked and discharge the power using the power button, then pull the video card, the memory, SATA (mark where they go 1st. And make sure you are using the regular sata and not the external sata or the raid sata ports) anything else on the mobo. Plug the power in and try to boot (it wont of course). Plug everything back in 1 by one. If you have 4 memory sticks, only plug 2 in. If you have any X4 or X1 cards, do NOT plug them in. ONLY plug in 1 set of memory, the OS drive, and the video card. Go into bios and do a 'check for any changes', then apply and exit once its done.
This will reset the bios what it has in memory/cache, and may 'find' the video card after.
You could also put both cards in the same system and boot to the old card, and see if windows even sees the other card (let it run a while, then go into nvidia settings and see if it shows both cards)
 
It really sounds like maybe 4 things:
Dead card.
Power supply is bad/not enough juice left to run that card/bad cable.
Bios settings are wrong, and it isnt booting the card correctly.
Not enough PCI-E lanes from you cpu/mobo to power everything on the system.

The video card has a shorted out component, the power supply is sensing the short and shutting down as soon as it trys to be powered on. 

Ryakkan
Just as a follow-up, I tested the graphics card in a friends computer with the same behavior. At this point I think when under load the power supply is somehow spiking or surging killing the power supply through the PCIe power ports. I'm getting a replacement one under warranty. Fingers and toes crossed.

The video card is shorted. It's junk. There have been lots and lots and lots of 900 series cards failing in this way recently. Replacing the power supply is futile.