What do you understand of efficiency?
Thickness (e.g. volume) is just one parameter of a profile. The big advantage of thickness is that you can put the supporting structure inside the aerodynamic surface, hence creating less drag. The second advantage is that a wing keeps its profile under load (more or less), hence it can be optimised. Long time the sahpe of sail was a result of mast stiffness, batten stiffness, sail cut and the skill of the sail trimmer. However this changed with time, as modern computer software can predict the deformation of a sail and hence it can be designed to have optimum shape under load.
However you can't look only at aerodynamics. The weight of the sail/wing and it center of gravity is a major contributor to overall efficiency. Clearly a wing is in disadvantage here.

Cheers,

Klaus