In a part winding motor you have two sets of stator windings, not 2 sets of rotor windings. A part winding start controls inrush current to some extent but not as well as some other starting methods. The second winding must be energized within about 2 seconds of the first. It is not a good method for controlling starting torque and current because of that and that winding resistance is rather low.
However, part winding start does allow use of 2 less expensive contactors instead of a single big contactor. The biggest single contactor that is not h#!!atiously expensive at 480 volts is 250 Horsepower. Even then, two 100 HP contactors will be less expensive than a 200 HP contactor.
Most air compressors that are 100 HP and larger use wye delta starting as that limits starting torque to 1/3 of across the line starting torque whcih saves a lot on gearbox cost. If you were to try to start a large compressor across the line you would actually break off gear teeth in the rotary screw compressor. Soft starters are starting to compete with wye delta starters.