I would be careful saying a run capacitor's main job is to correct power factor.
I'd rather say the capacitor shifts the phase of one of the windings so that the voltage across the winding is at 90° from the other winding. The effectivly makes it a two-phase motor (when loaded).
Now if the proper capacitor is used......then the currents through each of the windings, when fully loaded, should be equal....and under those conditions power factor will be close to 100%.
I agree 100%. In a nutshell, this is what I was trying to explain, but it did not come out as cleanly as you described. I was thinking about the phase angles and the reactance the capacitor adds to shift the angle.
Thanks for the input. My wordiness probably confused some readers.