Is there a way to check if current is passing through both windings, when wired up, with the multimeter? Since I replaced the capacitor it can't be that, so the problem might lie with the switch?
Thanks
At the moment, it won't matter much what originally caused the problem, since the fried winding is completely unreliable, even if it sorta works now (which I doubt, actually). Eventually, you will want to keep whatever it was from happening again.
Direct answer... set to ohms, connect to each end of the winding in question. The one with the capacitor, you need to connect to the wire after the capacitor, which will be one of the wires you connected the capacitor to.
We do not know the old capacitor is any good. So, yes, it could have been the capacitor. Currently, one winding is fried, so problems are quite likely.
As for the switch, it clearly connected power to the run winding, so it worked. For the type of motor it seems to be, power would have been connected to both, and the switch appears to have done that.
It could have been a stall of the motor, maybe a piece of lumber closed up on the blade, etc. Could have been a bad connection somewhere. Odds are the capacitor did not pass enough current to fry the winding, so only the "run" winding fried, but maybe the connection to the "capacitor winding" was bad. Can't tell from here.
I agree, the motor does not look that odd. Is there any sort of nameplate on it? Odd shaft sizes can be fixed with a lathe, or with an adaptor sleeve if too small. Make it fit the pulley.