The current isn’t there to support a slug of current like DC capacitors are, it’s there for a phase shift
Your air conditioner is fed by AC power which wiggles up and down. If you put a capacitor in series in AC power it shifts the timing of the wiggles so the output of the cap is some amount of the wiggle width off from the source
AC motors have windings/coils that we feed this wiggly power into. Really big 3-phase motors take 3 slightly offset wiggles which let them generate lots of torque from the start.
You only have a single wiggle phase in your home so we use the starting capacitor to generate a second phase and feed a second winding to give the motor an extra kick to help it get up to speed where the single phase can easily support ot
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