How deep do you want to go with troubleshooting vs just throwing parts at it?
https://www.amazon.com/gp/aw/d/B00NWGZ4XC/ref=mp_s_a_1_1?ie=UTF8&qid=1504477496&sr=8-1&pi=AC_SX236_SY340_FMwebp_QL65&keywords=clamp+on+amp+meter&dpPl=1&dpID=41sQt%2B0%2B-SL&ref=plSrchCan you safely use a meter? Some interesting thoughts I have.
You changed capacitor and it didn't start, but you can start it with a stick. The problem is still somewhere in the start circuit. Your motor is stuck in between 2 magnetic phases that are 180 degrees out. That's where the start circuit comes in. It gives a little starter magnetic field about 90 degrees out. Once rotation begins the start field is no longer needed since the momentum and phase change allow it to go from 1 field to the next field. That's why you can poke it with a stick and it takes off, your doing the job of the start circuit.
Obviously the motor isn't locked up or it wouldn't spin, however if you want to check the health of the main windings and bearings, you can check the amperage (with a clamp on meter like linked) . The motor plate will have an FLA value for full load amperage. Your amperage should be very close to this value. If it is over, there is some sort of mechanical resistance or winding failure. However I don't think that check is needed....
Something is stopping your start circuit from developing a magnetic field. If you changed the capacitor that is possibly ruled out. The only true way to rule out capacitor is to find a digital meter with capacitor check and check the old one, that's the 100% check.
The next thing to do is disconnect the motor start circuit windings and ohm the windings for the start circuit. Some digging around online should provide the desired measurements. It would be in a field service guide for your unit. If the windings are good, and the capacitor is good, there is something not turning on like a relay or bad connection in my opinion.
But I imagine you've already resolved the problem by now since it's hot out. If it was me, I would set my thermostat to a value so cold it can never get there, start the thing by hand and let it run constant without cutting off untill you have good time to dick with it lol.