Ken, I didn't notice the switch. From what I can see in that wiring, it looks like, possibly, the original switch shorted (always on, whether in the on or off position), then the hot wire was cut that supplied power to either it or the motor, and a toggle switch was inserted to function as the off/on switch... not the way I would have done it, as you'd be relying on a failed switch to carry the current needed to run the saw. When the contacts in it fail, they can melt/stick together, but with burns, cinders, or other less conductive material between them, and yes, this can keep the motor from getting enough voltage to run properly. It could even eventually burn the motor up.
If they bypassed the old switch entirely, as far as the hot(black) wire goes, it's possible the toggle switch they added wasn't rated for the current and it failed.
Either way, it's possible replacing the switch(s) with an OEM/replacement version may solve the whole problem.
One with some electrical experience could bypass the hot wires of both switches and if the balance of the wiring and the motor and capacitor was ok, it would run normally, thus pinpointing the switch as the problem, and be replaced.