No ordinay diagnostics can detect these problems. But your eye-balls can.
The diagnostic that actually can detect the bad Discrete GPU is a special diagnostic they used to keep at the Genius Bar. It takes nearly 1/2 Hour to run before reporting OK, but may report Failure in somewhat less time.
Macs before 2013 models use Apple Hardware Test (Not Apple Diagnostics). This is a World of Hurt because it was installed only on the original Boot Drive and on the release DVDs. Replacing/Erasing the boot drive REMOVES the diagnostic. Later versions of MacOS with System Integrity Protection make re-placing the diagnostic back into the System nearly impossible.
When you invoke Apple Diagnostics on pre-2013 Macs, it goes through the motions and always fails to load.