A single-phase motor will not usually start without a start winding because there is only one field providing push/pull in only one direction.
I think of it like a disabled cyclist with no knees. Initially, pushing down a pedal will only get him as far as the straight up/down pedal position. Give him a push and he’ll have enough inertia to carry him through that position and be able to do work pushing the pedal down again. A cyclist with working knees can use them to apply a (relatively small) force fore-and-aft at the up/down position and can get away without a push.
The motor start system uses a capacitor to cause the waveform to the start winding to lag behind the main waveform and give a push, analogous to the working knees, at an an angle to the main winding. Effectively the motor starts on 2 phases, dropping out the second phase once it is running fast enough to ride out the zero-torque orientation.