Most of the time, since the 2008 models, the problem is not the GPU.
They get way hot, and temperature cycling will damage the silicon over time, but the issue that caused the problems with these machines was not the chip itself but a tantalum capacitor in the power supply for the GPU, which was incorrectly specified and never designed to operate for so long in such a hot environment. Eventually it becomes leaky, causing the power supply to become unstable.
Since Apple don’t do component level repairs, they replaced the entire logic board. The real fix if you find someone who can do it costs 1p in parts.