The stator core flux is set by a combination of both the main and auxiliary winding currents in the stator, and the rotor mounted winding current. The auxiliary winding (with the capacitor load) therefore aids the excitation without supplying all of it.
So I think the purpose of the auxiliary winding feeding only a capacitor is that it provides a simple external way of adjusting the excitation. This is required when changing frequency, e.g. from 50Hz to 60Hz, the speed of the engine needs to be increased by 20% (to give the new frequency) which would also result in the voltage going up by 20%. So by reducing the capacitance, the excitation is reduced and the voltage is maintained at its previous level.
Incidentally, when large 3-phase synchronous generators are connected to unloaded lines (inherently capacitive) this can cause over-excitation i.e. voltage surges, hence reactors are connected to the line too. I for one can't provide an easy explanation of why capacitive current aids the stator core excitation, but I can accept that it happens.
The equivalent effect can also be seen in a synchronous motor. If the rotor is over-excited the motor draws capacitive current i.e. it has a positive power factor.