This is spiraling out of control for no good reason. ( and not because of the compressor you bought )

From what I recall of the pictures, the compressor in question had a very small pulley on the motor relative to the pump. Please measure the OD on both.

Where did you find a spec that this pump takes 2 HP? Is the motor data plate stamped 1/2 HP? Also please list the RPM of the motor.

The capacitor that failed, was it in a plastic or metal can? When start or run caps fail they do not damage the motor directly, any damage only occurs if the motor is trying to start and does not trip a breaker / thermal overload.

Does this compressor have an unloader? On smaller compressors this is a small ( 1/4" OD ) line that runs from the compressor head or tank check valve then to the pressure switch. This line is usually on the side of the pressure switch and is in addition to the tank pressure sensing line.

When the compressor is up to pressure and stops, the unloader line relieves pressure at the pump so the motor isn't starting under a high load. ( Another trick when dealing with limited power, add a manual valve to the unloader line and leave it open until the pump comes up to full speed. )

This can still be made to work, it is possible to run a " 2 HP " pump with 1/2 HP motor, it will just put less air because the pump speed will need to be lowered.