Your compressor should still be good. When you were testing the resistance between the three electrical connections to the compressor, you were actually measuring the resistance of the start winding in the compressor motor, the run winding in the compressor motor, and the start and run winding in series. The fact that your measured resistances of 4.6 and 7.1 ohms add up to very close to the third measured resistance of 11.1 ohms means that there are no shorts or an open circuit in your compressor motor, so it should still be OK.
The start relay on a refrigerator compressor motor simply cuts out the start winding once the motor has come up to speed. Years ago the way this was done was by connecting the coil in the start relay in series with the run winding of the compressor motor. When the compressor first started, the draw of current through the run winding of the compressor motor was high, and so there was high current through the relay's coil, and the magnetism generated by the relay's coil was enough to overcome the force of a spring and keep the start winding in the circuit. As the motor came up to speed, the current through the compressor motor's run winding diminished, and the result was that the current in the relay coil also diminished, resulting in the magnetism of the relay coil dropping and the spring pressure breaking the electrical connection to the start winding in the compressor motor. Thereafter, the compressor motor would continue to run on the run winding with the start winding disengaged. The "start device" you've ordered does exactly the same thing, but might not do it in the same way.
I found this web page on replacing the start relay on a Whirlpool/Kenmore side-by-side:
[FIXED] How to replace a (Whirlpool/Kenmore Model 106) Refrigerator Start Relay - ApplianceBlog.com - Appliance Repair Forums