I find the factor of Aragorn's leadership in both cases to be very interesting.
Not sure entirely what it means. It could be a way of bringing Aragorn's companionship with Frodo full circle or the signal of different stages of their relationship (the first time an increased load of responsibility for the injured Frodo, the second allowing Frodo to go his own way)
However, I'm not sure I find either of those ideas too convincing.
Probably, but for this purpose we come here.