I've had that happen with a magnetic switch. To fix it I cleaned the contacts. If your saw doesn't have a magnetic switch and just a regular switch you could try disconnecting or bypassing the switch. If the saw runs then the switch is the problem. If not the motor or motor overload switch is the problem.
I wonder if it could be a capacitor. Some motors have 2 caps, I believe one is for starting and one for running. I had a failure of my starting cap. The saw would start turning and then die. Replacing the one cap fixed it (~ Can $10). It's been fine for about a year since. In my case, there's like 'bubbles' in the motor case where the caps are, about 1" by 2 1/2".
Caution: Capacitors can store a charge kinda like a battery. You could get a shock even with the saw unplugged. Usually the terminals are shorted with a screwdriver or something with an insulated handle, just to be sure it won't bite.
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