But how does this explain the humming when the contactor is open? I suppose the start switch could be dirty and making a high-impedance circuit through the coil, causing it to hum but not quite get enough current to pull the contactor in, but geez, that would seem to require some precision dirtiness, wouldn't it?
I wonder if there is an alternative path for the coil current. Possible that the cord is frayed and one side is shorting to the chassis? Or the switch is slightly to large for its enclosure, allowing a contact to touch the chassis? That seems more likely to explain how bumping the saw triggered it - wiggled a wire against something, momentarily.
I'd carefully inspect all the wiring. Then, ohmmeter everything out to confirm nothing is shorted. Then, I'd hook up the switch to power WITHOUT the rest of the saw, and see if it hums (it shouldn't!). Then, connect the motor (ideally not using the wiring in the saw). Etc. I bet you'll find a wire that's been rubbed through, somewhere.