So long as you have a supply to the motor it should run, might be worth trying to wire it directly bypassing the pressure switch to prove the motor is good...
Got to say, if it was struggling to make pressure this would tend to point to the pressure switch or compressor pump itself, rather than the motor.
If the motor was at fault it would keep running till the pressure switch told it otherwise, and would just take longer to build pressure...
Saying that, if the cap is faulty the motor may not be producing as much grunt, which would mean it would work with a lesser load but then could stall out and stop before the pressure switch turned off. The motor would just then get hot until the termal cutout switched it off.
I am guessing that you are trying to start the compressor on an uncharged tank, and starting with a tank already pressurized will cause a weak/faulty motor to struggle.
You say the motor is buzzing...so that would tend to suggest that the pressure switch is telling it to start...which again points back to the motor.
Check the windings, try the motor with the belt off...if still no good change the cap