From your symptoms I guessed that it was a capacitor start, capacitor run motor. If you look at the attached sketch, the way the motor is reversed is that the two leads for the start winding are switched by the directional switch. Sometimes the run winding is switched but that is rare. What I think is happening is that in the direction that operates normally, the start and run windings are connected. But in the other direction one of the leads for the start winding is not connected which leads to a no start and poor peformance. My money is still on your reversing switch.

BTW, shorting a capacitor doesn't hurt it. In fact, if you are working on something with large capacitors it's normal procedure to discharge them rather than get knocked on your butt.