The symptom of running in either direction if a manual spin up is used at start is a classic indication the start winding did not connect to the capacitor.

 

Would that be cap polarity or something else? There’s no markings on the of cap and both wires are blue.

 

If its a start only cap, not start/run, then it could be bi-polar electrolytic, either way, polarity does not matter.
The symptom would be either bad capacitor, or no connection at start up.

 

A capacitor run induction motor will be weak if the capacitor is not large enough. I had that problem with a bench grinder. It would not spin up, and it got hot. A capacitor with actually the rated value actually present was the fix.