Not sure that the reluctance of Aragorn worked well for me, but then again I had read the books many times before seeing the movie, and so knew that eventually Aragorn would take the throne - no matter what else PJ changed
.
To me the turning point for Aragorn is when he speaks with the little boy in Helm's Deep regarding the boy's sword. At this point I think that he realizes that it is up to him to not only physically enter the battle but also to take more of a leadership role, as it is no longer just about himself, Legolas and Gimli. Prior to the battle at Helm's Deep, Aragorn is consumed with trying to be the
anti-Isildur, making up for his ancestor's folly.
From the movie we are to believe that Theoden would have failed against the Uruk Hai of Saruman, and if not for the assertion of Aragorn to 'ride out at dawn,' then they would have been doomed. After this, Aragorn is on a roll, leading the Army of the Dead then the Army of Free Folk.
Politically he would have gained the support of Eomer from saving Rohan, the support (even submission) of the Gondorians after saving Minas Tirith using the AotD, and the allegiance of Faramir, the Elves (due to kinship and they knowing his ancestory; plus they were leaving anyway), and the Hobbits. Who would be left to contest his ascension to the throne?