It is often difficult to tell whether CPUs are suffering...older socket style CPUs can have memory chips on the top surface touched and damaged...I have read about problems after heat transfer compound was added and then spread to places it should not have gone.
Slot loaded CPUs are very vulnerable to damage if the machine is dropped...[i think everything is

]
CPUs can meltdown fast...and fans that arent working properly can damage the capacitors and so on supporting the CPU making one suspect the CPU...swollen or leaking capacitors are a signal of this...but some people miss it.
But as mentioned by others, the only way to test anything for certain is by swapping.
If you are lucky enough to find a tester for a CPU, then it may pick something up...but I do not know if such a tester exists...
Swapping is the key
Roley
Black Frog Tech