If you see the left side of the Planck which the ball touches, it seems so that as we vary $\theta$, the contribution of the force from that point of contact drops to zero. I have marked the point of interest in paint:
As we reduce $\theta$ , we see both the edges touching the sphere contribute to supporting it's weight. But, as we increase $\theta$ it drops off. It is intuitive to understand for me, but I can't give a precise reason why as to this happens. Hence, the question:
For what exact reason does the normal force shift to the right edge as we vary $\theta$?