Is there air pressure in the tank when trying this? If no pressure then your motor run capacitor might be bad ( they generally lose capacitance, and when it drops below 80% of the rating the motor generally will not start), or the pressure switch might be defective, though what often happens is the unloader valve, which is the thin plastic pipe that goes to the side brass connector of the switch direct from the compressor piston outlet, is faulty. Another common issue is the non return valve on the connection to the tank starts to leak, and then the compressor has back pressure and will not be able to start as the motor does not have enough torque.
These motors rarely have centrifugal start switches, they are almost all permanent split capacitor type as the motor generally is starting into an unloaded compressor and can get enough momentum into the compressor so that when it does build up enough pressure to pump it will be at full torque and will keep on running. Then when set pressure is reached the switch both disconnects power and also dumps the air in the compressor cylinders via the unloader valve, with the non return valve on the air reservoir keeping the tank from losing pressure via this valve. That is why you must use the compressor switch to power it on and off, as otherwise it might try to start on a full pressure line and will stall.