A couple of important distinctions here:
- Angle of Attack (AoA) is the local angle of the blade profile relative to the airflow.
- Blade pitch is the local angle of the blade profile relative to the propeller rotation axis
- Variable pitch propellers can offset the whole blade with an additional angle. The shape of the blade will remain constant.
Picture above is from this link, which contains a good explanation on propeller blade shapes. It shows the AoA, with the two velocity components Relative Wind and Rotational Velocity. And the Rotational Velocity increases with increasing local blade radius. So in order to have a constant local AoA at every point of the blade, the local blade pitch must be a (linear) function of distance from the centre. Which can be seen in all propeller designs, old and new, fixed blade or variable pitch.
The term Variable Pitch may cause some confusion. All propellers have a local profile pitch angle relative to the spinning axis that changes with local radius, like in the pic above. But Variable Pitch means: the whole blade can increase its pitch angle, to keep the AoA constant with increasing airspeed.