Let's try to eliminate things one at a time.

First clean the centrifugal switch contacts and see if that helps. A fine grit metal sand paper works well. Sometimes the contacts can corrode and cause intermittent problems. Because the motor doesn't start in high speed, this could be part of the problem.

Next, if you have a multi-meter you can check the capacitors:

1 Fully discharge capacitors with a screw driver across the capacitor terminals
2 Remove capacitor(s)
3 Set ohm meter on lowest resistance setting
4 Measure resistance across terminals - Should first read close to 0 ohms and then steadily climb to infinity. If it is shorted or always open, then the capacitor is bad.

If both capacitors check out and the centrifugal switch has been cleaned, then it could be a short in the main winding. The low speed winding is separate from the main winding so you can get thermal overload one and not the other. Unfortunately, that would not be something that could be easily fixed.