Oh wow, that's far worse than I had pictured in my head.
You mentioned the GPU works as long as the driver isn't loaded. I have seen some strange things happen. There is still a possibility this is a just a software issue. I have seen GPU overclocking profiles/conflicting apps that both manage the GPU loading @ startup that kill the display as soon as it boots. Can you boot into safe mode and remove all GPU overclocking apps. If they ask if you want to remove all settings make sure you remove them, that's the main fix right there. Use MSCONFIG to control what is starting up and clean house.
If it still doesn't work then you may just have a corrupt driver. At this point you should probably give DDU a try. Download the latest display driver uninstaller and I would recommend running it in safe mode.
After that process is done you should be able to just download and install the latest NVIDIA driver again.
If all that fails then you might indeed have a bad GPU however the very last check is to install the GPU in another computer as a test. It is easy and quick enough to try.
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Xeon E5-1680 V2 (IVY-E) Inside