Changing flap setting changes stalling speed because it changes the MAXIMUM lift coefficient. However, with plain and split flaps, changing flap setting doesn't change the instantaneous lift coefficient
Aaaaaaaaaah. But it does. Let us assume we are flying at a given speed with no flap extended. The wing will have an angle of attack of X° and be operating at a lift coefficient of Y. If you maintain the same speed and extend flap the angle of attack of that portion of the wing that is flapped is now (X+A)° with a corresponding increase in the lift coefficient. The unflapped portion of the wing, because the flapped portion is now producing a greater percentage of the required lift, has a reduced angle of attack and a lower lift coefficient and produces a reduced percentage of the total lift. As a pilot you notice the effect that when you lower flap you have to lower the nose to maintain the status quo. A diagram of the span loading of a flapped wing will show a discontinuity at the outboard end of the flap, compared to the smooth elliptical shape of an unflapped wing.