Reply to Amen Brother
Thanks for your prompt and thoughtful reply. 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. Here is why. Lift coefficient is equal to lift divided by dynamic pressure and wing planform area. (Lift is equal to aircraft weight times instantaneous load factor. Dynamic pressure is indicated airspeed squared, times half standard air density.)
With plain flaps and split flaps, changing flap setting changes none of these things - no change in weight, load factor, indicated airspeed or wing planform area. Consequently changing flap setting doesn't change instantaneous lift coefficient. In the minute or two after changing flap setting there is usually a significant change in airspeed that brings about an equally significant change in lift coefficient, but it is the change in airspeed doing that, not the change in flap setting.
With Fowler flaps, there is a change in wing planform area and hence a change in lift coefficient. If extending Fowler flaps increases wing planform area by 10% then lift coefficient will decrease by 10% as the flaps are extending. However, the subsequent reduction in airspeed will cause the lift coefficient to increase to its original value and higher.
My questions asking why designers don't produce airplanes with fixed trailing-edge flaps, and why pilots don't fly all the way to the destination with takeoff flap, to minimise trailing vortices and induced drag were rhetorical questions.