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 points. The tangent points are not on a plane but on a "tube in hyperbolic space" / "binodal cyclide" / "inversed circular cone" (help ... what's the standard name for this surface?) .