Remember, a propeller has a non-constant angle-of-attack of the airfoil section when you go from root to tip. I was searching on the web for lift vs AOA experimental data, and someone found that 5 degrees to 25 degrees gives the optimal lift-over-drag ratio for a typical airfoil. So while increasing propeller pitch can increase lift toward the tip, the section toward the root will approach stall and generate very little lift; hence the lift generated by the entire prop is not too much different.

Seanychen -

I would just like to comment about this piece of technical info....lift is the product to airspeed and AOA. You prop has a higher AOA at the root because its airspeed is slower than the tip. The net force (thrust) is the same at the root of the prop as it is at the tip. The tip is not near a stall because it does not and will not exceed its CAOA (critical AOA), which is the only time an airfoil will stall.