IMHO, once R has signaled for and completed a fair catch, his team is awarded a new series. What his team does after that is entirely up to them. They can choose a free kick, choose to snap the ball, and/or choose to attempt a scrimmage kick. In my mind, the basic fundamental is clear: "If R is the first to touch a scrimmage kick beyond the neutral zone, a new series will be awarded to the team in possession at the end of the down."