You are right when you say that the way that things are portrayed is different between the books and movie.
Actually i think its covered pretty nicely in
Aragorn's decisions from what I read. In the movie they leave out that whole part with Imrahil and the House of Healing. I think that part really helped gain the respect and trust of the people. Through that whole time between the battle at Pelennor Fields and the time when they march on the Black Gate Aragorn refuses to enter the city as the king. He is recognized as the Captain of the Dunedain of Arnor for the time being and remains in a tent outside the city. So when they (Aragorn, Gandalf, Imrahil, Eomer) lead the armies against the Black Gate and they are victorious, and he is finally revealed to them as Aragorn, the rightful king, they realize that this guy that they trusted in battle, as a result of his skill, is going to be their king. So by the time he actually take the thrown the people know that he can be a leader because of how he lead them to victory over Sauron, and then they are more willing to accept him.
I asked and we can continue to compare this to the books as long as it doesnt turn into just a book discussion.