There are propellers with different aerofoil sections.
For example, the propeller attached to the Rolls Royce engine kept in my graduate college had symmetric aerofoil sections roughly about 25 % and smoothly varying to bottom flat airfoil sections at the rest.
The aerofoil profile selections are based on the performance requirement, structural rigidity, and modes of operation.
You can go for any aerofoil sections with high ($C_l / C_d$) but, in the end, it should be structurally rigid to produce high thrust at high RPM. When you have variable pitch propellers you don't need different aerofoil sections since the required performance achieved by adequately rotating the blade.
In my point of view having the same aerofoil sections will be advantageous in the below-given ways.
- Matching the aerodynamic centres of the same aerofoils sections would be easier
for example, NACA 0006 have the $a.c$ at $\sim c/4$ and NACA 2412 have at 0.3$c$, then locations of $a.c$'s will not be smooth along the span of the propeller. This would be critical in structural stress distribution on the propeller.
- Downwash of the propeller sections will have smooth variations.
For similar sections, the downwash induced velocity component will be predictable and the twist of the blades could be compensated easily while designing. And the twist will be smooth for similar profiles from the structural point of view.
Hope this helps.