Why the torque of single-phase(split-phase induction motor) is not zero?
Once the motor begins to spin up, the mechanical momentum of the rotor, now acting as a flywheel, progressively becomes the primary source of the torque. in the end the torque is effectively zero at no load (assuming no friction at all), but is greater depending on the actual load. Under load, and once rotating, the torque formerly contributed by the electrical phase angle between the main and starting coils is progressively replaced by the "slip" angle, which is the number of mechanical degrees behind "0" between the rotor position and the crest of the current waveform feeding the stator coils. That end state occurs when the rpm is in sync with the supply frequency. So the starting coil contribution no longer matters at that point, because the rotor is receiving the majority of its added torque based on the purely positional slip angle, which is much greater than the contribution of the starting coil, which as you said can now be switched out. 

I do apologize that I can not illustrate mathematically for you what is happening between the start up, transitional, and end states, as it has been over 40 years for me since I've studied how various AC motors work, and I've long forgotten most of those details. But maybe the explanation I'm offering will at least help the operation make sense.