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So, you got a winner ;) That's why usual methods such as SMC reset and other tricks gave no result with your issue. There must be some protection in firmware which puts your mac to sleep/low power mode when CPU temp is reaching critical value. I bet that after replacing failed fan your issue will be gone.
 
Dust bunny

I had similar symptoms except iStat showed my fans working but with the CPU fan showing at 3200rpm when idle speed should be closer to 1200rpm. At least that showed me that the fans were spinning. Also, the GPU sensors were quite high at around 65 degrees celcius. It would sleep more quickly as I intensified the duty load (ran 6 videos at the same time). So I needed to find out why my CPU fan idle speed and GPU temps were so high.

I took it apart and found it caked with dust bunnies everywhere. Those things really do breed like rabbits! So I removed the casing and LCD and used a compressor to blow out the fans and heat sinks. With the bunnies removed, the CPU fan now idles at 1200rpm and my GPU sensors are in the 38-40 degree range. No more sleeping. :)
 
This may not apply to the problem from nearly 10 years ago, but keep anything magnetic away from the laptop, including cell phones and smart watches. I am sometimes able to make mine go to sleep by putting my 1st gen apple watch near the left side of the macbook pro.
 
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The magnet was exactly what was happening to me. In my case, I had the AirPods case in my hand while in a video conference and didn't realize that I kept putting it near the middle front of my MacBook Pro, it would put it to sleep. Ha ha.
 
I had this issue with my old MacBook Pro. At one point out had a broken fan and would go to sleep whenever the computer was too hot and the remaining fan couldn't cool it. My temporary solution (until fixing the fan) was putting a huge fan behind it whenever I played a game or did something intensive. If that helps anyone
 
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