How to determine if the graphics card is at fault on an iMac?
You should try safe mode. Safe boot: Shutdown your machine. Hold down the shift key. Poweron. The boot up will take longer than normal because the filesystem on the startup drive is being checked and repaired as needed. All about safe mode including what features and apps safe boot leaves out. Safe boot uses a software driver instead of using your machines video hardware. http://support.apple.com/kb/ht1455
If the machine works in safe mode, you can set up your machine to run the safe mode video drivers in normal mode.
Runs OK in safe mode...
Running in Safe mode leaves out some video drivers. Which results in your machine not using advanced video hardware. As luck would have it, you can run the safe mode video drivers in normal mode.
Here is how: https://discussions.apple.com/message/16057567#16057567
Look through the above thread. See the second page. You don't have to read through the first page. Just go to the part where I try a solution that works.
On Sierra and beyond, you need to turn off SIP, System Integrity Protection before making the changes. You should turn SIP back on after you're done making the changes. SIP prevents alteration to system files.
https://www.macworld.co.uk/how-to/mac/how-turn-off-mac-os-x-system-integrity-protection-rootless-3638975/
Using the safe mode video driver disables accelerated graphics.
Multiple GPU restarts and resets on 2009 and 2010 era Macs is almost certainly GPU failures. If you wanted to get two or three scrap Macs and teach your users repair / see if they can learn how to scavenge parts - that would be a wonderful education for low cost.
Here is an amazing (to me) story of a DIY reflow of the GPU to salvage the existing hardware if the GPU has simply lost electrical connection with the hundreds of pins that connect it to the system.
- https://apple.stackexchange.com/a/257567/168832
If you want a stable computing experience, I would get a $20-30 raspberry pi or even an Arduino and work up from the low end with them.
There always will be some fraction of Macs that run forever, but the longevity of these early model Intel Macs isn’t as long as the PPC predecessors. I’d go for a modern computer and iOS device for games / messaging to get much more bang for your buck and less time maintaining the hardware.