Experimenting with PC builds may teach you a lot of things but also may damage some hardware. It is a rare problem to have your PC fan falling by itself. Maybe it wasn’t correctly placed. If there was a problem with the fan while you were playing, maybe there was an overheating issue and that is why the PC had to shut down.
Could you give me some more details about that CPU? What do you mean when you say you have removed it and it was running?
I would recommend you to apply some thermal paste and search for a new CPU fan. Maybe the best thing to do is to consult with a technician if it will be possible to try with another fan. There is no point buying a new one and then realize that the CPU has a problem too.
Hope this helps