According to the discussion in Coxeter (1968), the tangent points lie asymptotically on a concho-spiral, so the distribution is not uniform on the sphere, but is uniform on a circle.
By the way, the circle is in general not a great circle. In fact, the process can be bidirectional, so there will be two accumulation point. The tangent points are on a "binodal cyclide", not on a plane.