Has my GPU died?

Hello Reddit.

I think my graphics card might be dead but I am not sure. I will try to explain my problem as detailed as possible.

My graphics card is a Radeon HD 7870, about 4 years old. Last week I bought the first episode of the new Hitman game. I played a few hours that night and had no issues. The next time I'm playing this game I'm about 2 hours in and suddenly a have a general system crash: my xbox 360 controller is stuck on 100% rumble, my monitor has a sort of brownish-green colour and my speakers produce that typical crashed system-audio (sounds like "RRRRRRRRRRRRRRR"). So I press the reset button on my PC, the PC reboots and loads windows and I get a black screen. The monitor flicks off and back on to show me a black screen. At this point the entire PC has crashed again: no video, the keyboard is not responsive (pressing caps-lock or num-lock doesn't light up the indicator button and the reset button on the case doesn't do anything. This only happens AFTER the screen flicks off and on again, before this point the keyboard and reset button are responsive.

The computer does however boot up without any problem in safe mode. I google a bit and figure it is probably the GPU that's causing the problem so I uninstall the GPU drivers and the PC now boots up normally (but without GPU drivers). I reinstall the drivers and the problem comes back. Installing older drivers (I think around 14.3 and down) gives me a glitchy image with stripes for about a minute before BSOD'ing into a reboot.

At this point, I open up my case to observe the GPU and I can see that one of the fans is not spinning when I power my computer on (the card was also FILLED with dust, I was actually kind of disgusting with myself there ..)

I figure that my card overheated and died due to heavy load, build-up of dust which prevented airflow and general wear from years-long usage. Is there anything I can do to confirm that the GPU is dead other than putting it into another pc? I've also thought it might be the PSU that's broken. By drawing power that the PSU can't provide, the GPU causes a general system crash when it's drivers try to make the card work. The fact that one of the two fans on the GPU don't work make me doubt this tho ..