After decades in repair, you ignore any lines like "not heat". You just run into heat issues where a chip or board that has become heat sensitive. That is, when new a board can run at extend temps. Years pass and it won't. It's that simple.
The build looks very old. You have to ignore the not heat then treat it for new heatsink compound and cleaning. A fan test is made than you leave the cover off and see if it stands up to testing.
The shutdown on high game settings often points to GPU, PSU or motherboard issues. Which part can you change first?
https://www.newegg.com/Product/Product.aspx?Item=N82E16817182200 has mixed reviews but it seems to be a nice enough single rail so it's lower on my list. Try GPU then motherboard.
I just recently started having this issue where my computer shuts off completely when trying to load Leauge of Legends (it happens at other times but mostly while in game or right when pre game lobby is over). Once the computer shuts off and when i go to turn it back on it normally shuts down right after windows chimes. After that it will load just to desktop just fine. I recently just installed a fresh version of windows 7 ultimate, I have made sure to download all potential drivers and update all windows files. I troubleshot while in a custom game by lowering my in game setttings from medium to low and when doing so my computer instantly shut down after going from medium high to low setting. I'd like to think it is the gpu but from all the forums ive been reading it could be a whole list of different issues. my specs are
Ta970 5.2 mobo
nvidia gtx 560 ti
Corsair Vengeance Blue 4GBx2
samsung ssd 840 (THE OS IS ON THIS)