first thing i always look at with single phase motors is the capacitors, they only have a limited lifespan and can often result in a burnt out motor if they are not changed.
if they are on their way out and then they generally don't have a lot of umph to kick the motor over, a slightly tight spot would be enough to stop things from starting up.
could also be the centrifugal switch though, if it get stuck in the run position then it wont switch the start cap in and the motor will just hum, by moving it a bit it might be resetting things and letting it start up.
try removing the belt and testing the motor on its own some more,might give an indication if its a tight spot on the machine stopping things and you should be able to hear the centrifugal switch click in and out.
tbh if i have any doubts i normally just slap in some new capacitors, nine times out of ten that cures things