its' not a GPU issue. its a logic board issue.
the other day i was working on a first generation (ATI X1600) mbp with the same exact problem. But this one would boot once out of every 10 restarts, and sometimes freeze at the desktop. The rest of the times it would just stay at a black screen with a solid LED, hard drive and fans running, but you can tell it wasn't doing a thing (caps lock LED not responsive). It was a vegetable.
swapped out RAM, RAM slots, SMC reset, etc. nothing worked. didn't seem like it wanted to do a firmware restoration, or update the firmware (when i got it to boot).