Found my can of compressed air. Cleaned my GPU. Wasn't very dusty, probably because the fans and chips were pointing downwards. Switched the computer back on, GPU still didn't work, kept giving a noisy fan sound and going blank whenever I tried to put it under load.
Decided it might be the power supply unit. Took it out, cleaned it off with compressed air. Discovered a dust filter at the bottom of the case that hasn't been cleaned in 8 years. Washed it. Put everything back together and the $3500 GPU started working fine. Problem solved.