Single phase AC compressors often fail because of the start cap.
I saw a demonstration with the start winding open. The motor is unable to start even under no load. Given a spin, the motor will start and run fine without a load. If the leads that go to the start cap are shorted, the motor puts out a small torque at low speed and is able to start if there isn't too much resistance. The start cap increases the low speed torque. The cap shifts the phase of the start winding causing the motor to act sort of like a 2 phase motor. That shift increases the torque at low speeds. Not as good as a 2 or 3 phase motor, but close enough.
A few years ago, the same happened to the A/C at my work. The HVAC tech replaced the cap and a fuse that blew of 2 one for each line going to the AC compressor. A year later, the AC went out again. After some troubleshooting, the cap was good, but the fuse that wasn't blown had blown. I assume that it was damaged when the A/C motor blew the other fuse and eventually failed. My friend decided to use a piece of 18 gauge wire to bypass the fuse.

At least there was the other fuse there in case of a locked motor condition. The wire actually worked for long enough for me to get a replacement fuse, there wasn't any noticeable insulation damage.
Nice pictures of the PF correction caps.