[Gurthang]

It seems to me that it was more a matter of priorities that kept him away from the city. In the movies, he did not believe that the Ring could go to Gondor and come out again, unless to the hands of Sauron. So he felt compelled to keep the One far from Minas Tirith. After he left Frodo, and the Ring, saving Merry and Pippin seemed to be a more urgent task. Hence, his reason for again steering away from the city.

Also, I'm not sure he thought the timing was right to claim the throne. If he did make a claim, there would undoubtedly be fighting between him and Denethor, since the Steward was disinclined to give up his rule. There would also be a question if anyone would support him at all. Either way, it would leave the city weaker right before an impending war.

One other point. He may have been thinking that he would not survive the War of the Ring. So any claim to the thone, whether successful or unsuccessful, would be very short lived.