It could possibly be that the run winding is somehow not in the circuit. When the motor is first applied with power, the electrolytic capacitor (the one that was bad and replaced), the stationary switch (white looking device in the center of the video), and the start winding are in the circuit with the run winding. After the motor shaft begins to rotate, the governor (the metallic device that the switch sits on) slings in by centrifugal force causing the switch contacts to go open and the electrolytic capacitor is now out of the circuit (no current flow). If the run winding is not in the circuit the motor would start but as soon as governor slings in there would be no current flow to the motor,due to both windings being open, the motor would slow down causing the governor to return back to its original state causing the switch to close and capacitor to once again return to circuit of the starting winding and the motor would begin to accelerate again. If you continue to run this motor like it is you will either blow the capacitor again and/or damage the switch in the process due to the continuous cycling of the switch.
Just two thoughts come to mind either the run winding is not in the circuit or possibly the run capacitor (the capacitor that is under the cover on top of the motor) is bad. Checking the run capacitor would be the easiest to look at before sending in for repair.
If the run winding is open you might check to make sure that there is a circuit through both sides of the overload(the black device that should be above the switch/governor)
