Thanks again Prof. Hwang.
The computation of the projection probability for linearly polarized state of light using Malus Law is mathematical rather than based on any logical physical law including the wave aspect that you tried to explain. It has been shown that Local hidden variables model that tried to explain the projection probabilities of two linear polarizers does not work (see Von Neumann Impossibility proof). The claim by Quantum Mechanics is its strength in predicting an outcome based purely on mathematics-- in this case it happen to be the cosine square and not any other trigonometrical functions.
Could it be that in the case of circular polarization, Quantum Mechanics has over simplified the problem by making the assumption that there exist a distinct right or left circularly polarized state that in fact does not exist. Could this be the explanation why we do not get a consistent result most of the time?