PC startup issue: Can a video card short damage a motherboard?

My PC was working fine until the last time I used it around 2 years ago. When I took it out of hibernation a few days ago, it would not start up. Upon pressing the power button the first time, all the leds came on for a brief moment and then went out. After that, pressing the power button had no effect at all unless I turned off the power supply or disconnected the power cable for ~10 seconds and reconnected it. Then the same thing would happen; one time only, the leds would flicker on for a split second and then nothing.
I began disconnecting connectors and found that the computer would fire up with the ATX12V connector disconnected. I hunted for a short and found a literal bug on the video card (see linked pic).

Bug

I removed the bug and reconnected everything. The computer now starts up. All leds and case fans run, but I get no video. Also, the fan on the video card pulses on and off approximately once a second.
This pulsing of the fan does not take place if the ATX12V connector is disconnected; it runs at a consistent speed, but still no video.

I tried the video card in another PC and it works fine.
I disconnected everything from ram to hard drives to usb ports and I replaced the BIOS battery, but the fan pulsing persists unless the ATX12V connector is disconnected. Only then does the fan run normally, but of course, no video still.
I suspect a failed MOSFET at this point, but that usually happens due to overheating

Is it possible for a MOSFET to overheat instantly from stone cold in a single voltage spike?
Any other suggestions or observations would be appreciated.

The motherboard is a Gigabyte GA 970A UD3