Ahhh, so that spring like thing, sat on top, between the two coils, could be a thermostat, that expands, and pushes the switch off, when it gets to hot, killing the motor.
Sort of like a lock out, you need to reset it once its triggered. I suppose it's got gummed up and failed.
I'm just guessing this mind.
Some kind of thermal cutout switch as a replacement then? There must be something on the market. I guess knowing what temperature it needs to cut out at would be the thing.
Could be worth quizzing some of these motor rebuilders. See what they say.
I'd probably just cross the wires and carry on. What's the chances of the motor overheating?
Of course if it's running all day, unsupervised, then that's a different story, but just in a private workshop, it may well be turned off unless it's being used.