May I add one other simple way of looking at this.
Firstly, what you need to remember about a slipball is that in steady state conditions there is only one force acting upon it - gravity. However, the airframe has both gravity AND aerodynamic forces acting upon it.
In a steady turn with no sideslip, if we view the aircraft from behind (YZ plane) there are only 2 forces present, gravity and lift. (Note that the radial force which maintains the turning flightpath of the aircraft, the centripetal force, is actually just the horizontal component of lift). For the slipball to be deflected, there must be different lateral forces (along the Y or lateral axis of the aircraft) acting on the airframe and the slipball. Lift is perpendicular to the lateral axis and so has no component along it. Gravity acts on both the airframe and the slip ball. Therefore, the slip ball must remain centered. QED.