AC Heat pump 16,000 btu cruisair which I have uses a start and a run capacitor, and a potential relay. Any one of those 3 goes bad it will not start and run.
You can also ohm out the compressor.
I think a lot of compressors are condemned by HVAC folk that want to sell you a new system when it is something else wrong.
For my F series compressor, it is like 0.8 ohms run windings, and 8 ohms start windings. A dead short does mean it is bad.
I have had to replace all 3 of those at various times. When replacing start and run capacitors, I jumped up to the 440 volt ones as they are going to last longer. Same capacitance, with a higher voltage rating means less likely to explode and fail.