Locate the onboard diagnostic LED’s in the bottom left corner covered by some black tape
Here's a breakdown of the onboard diagnostic LED's:
LED 1 - Indicates that the trickle voltage from the power supply is detected by the main logic board. This LED will remain ON while the iMac is connected to the AC power. The LED will remain on even when the computer has been shut down or put to sleep. The LED will turn off only if the AC power is disconnected or the power supply is faulty.
LED 2 - Indicates that the main logic board has detected proper power from the power supply when the computer is turned on. This LED will be ON when the computer is turned on and the power supply is working correctly.
LED 3 - Indicates that the computer and the video card are communicating. This LED will be ON when the computer is communicating properly with the video card. If LEDs 1 and 2 are ON and you heard the startup sound, but LED 3 is OFF, then the video card might be installed incorrectly or need replacement.
LED 4 - Indicates that the computer and the LCD display panel are communicating. This LED will be ON when the computer is turned on and video signal is being generated. If the LED is ON and there is no image on the LCD display panel, the LCD display panel or inverter might be installed incorrectly or need replacement.
What are the LED’s telling you? Let us know too!
I think you've narrowed the issue to the GPU board.
The fact About This Mac and preferences shows the display is present doesn't get into the GPU and if it has a problem.
This series often encountered issues with the GPU. The best way to explain this is how fast both hardware and software has evolved! Here at the time the systems being designed around 2008 it was not even in anyones head how the graphical world would have evolved in a few short years. So while the designers of the GPU's where pushing things they where eclipsed by the software designers!
Now let's look at a horse which is scared thinking something is after it. A horse will run to its death!
Thats what happens with older GPU's! They run until they die when pushed to the limit.
Its time to try cleaning the old thermal paste off it and apply a fresh coat iMac Intel 27" (Late 2009 or Mid 2010) Graphics Card Replacement
You’ll also need to replace the thermal pads on the VRAM chips Thermal Pads
Lets see if that gets you going if not either replacing the GPU board or baking the board which can often give you a bit more life.