My card died
I got a dxgkrnl.sys BSOD and it was saving the minidump. Once it's done, it's supposed to reboot, but it did not. When I rebooted the PC, the monitors had signal, but there was no picture.
Installing my 780Ti to confirm it's the card that is to blame and will contact EVGA RMA.
//Edit: looks to be something else - 780Ti has the same issue. Motherboard (ASUS Rampage IV BLACK edition) shows code 62, whatever that is. Off to do research
Ok, after 6 hours, the conclusion is that the motherboard derped and no hardware failure occurred. For some reason, my motherboard decided to reinstall iRog1 and iRog2 (whatever those are - tbh no idea). However, it appears DisplayPort gets activated at a later point, so all I was seeing was a black screen, but it was actually doing stuff in the background. Followed a suggestion from an older thread to connect a monitor to the DVI port instead and everything became clear. Just in case, also ran Memtest86 on the memory.
Additionally, I moved Hybrid's fan to the motherboard fan header, but I don't really have a way of controlling it at the moment (SpeedFan doesn't see a fan there). For now, I left it running in a Quiet profile (configured in BIOS).
That said, I did get dxgkrnl.sys (DirectX Graphics Kernel) BSOD that started all of this. There was no overclock at the time, so I guess I should just blame this on the NVidia drivers?