seems to be a misconception that all single phase motors are run with a cap. There are some that are single phase without the need for a capacitor, that incorporate a centrifugal switch. the motor has 2 windings a start winding and a run winding. the C/S is in line with the start motor as soon as the motor gets up to speed the C/S opens.
Err, suggest you learn a bit more about motors.
TOther than some fairly obscure ones you aren't likely to see, the only single phase motors that don't use a cap are shaded pole - and they are only used for very small motors (small fans etc) because of their low efficiency/high losses (= heat production).
On motors with the centrifugal switch you mention, you will find a capacitor. The switch is there to switch out the start winding (and it's capacitor) once the motor is up to speed.
These days manufacturers tend to prefer capacitor run motors which are cheaper to produce (no centrifugal switch, and the "start" winding produces some of the field all the time) - but they have a poor starting torque curve compared to cap start motors.
The key thing is that you need a means of creating an "out of phase" current so as to get a magnetic field with some rotational element to it. If you simply connect 2 windings to the same single phase, then all you'll get is a field that flips back and forth with no rotation.
On most single phase motors, the capacitor is used to introduce a phase shift to the current in one winding. On a shaded pole motor, the shorted turn (the shading ring) does this by altering the magnetic characteristics of part of the stator.
Without such a phase shift, you have to spin the motor before applying power - rather impractical for most applications.