[The Squatter of Amon Rûdh]

Well, not entirely. Great Britain officially recognised the independence of the United States in 1783, so there'd be no legal precedent for a British monarch to claim any rights of sovereignty there.

A closer parallel would be for a direct descendent of James II to turn up and claim to be the rightful king of England. Although the ancestry would be right, such an action would ignore the events of 1688 that removed the Stuarts from power, and every constitutional change since then.

Aragorn's situation is very similar to that of Arvedui, but his claim is reinforced by his work on Gondor's behalf in the War of the Ring. He is well aware of the constitutional issues of claiming the kingship, which is why he refuses to do so when first he comes to Minas Tirith. He may use the banner of the Kings of Gondor in battle, where it can be a boost to morale, but when he enters the city, he does so as a private individual. In doing this he demonstrates a patience and flair for diplomacy which, combined with his other kingly qualities, would help his case immensely. Also he does everything he can to do due honour to the house of the Stewards, following up his initial tact by confirming Faramir, Denethor's rightful heir, as steward.

Although I agree with Mister Underhill that Aragorn's huge popularity would have played a part, I'm sure that the shadow of the Kin-strife lay heavily on his mind, particularly since he had already met Denethor and knew how he would view any attempt to establish a king in Gondor. Given the precarious position of the country at the time of Aragorn's return, he would probably have waited until after the war had been fought before making a bid for his inheritance, had Denethor lived and remained sane.

It's interesting to note that in deliberately breaking Denethor's mind, Sauron does much to aid the return of the king. Although his intent is to weaken the leadership of Gondor, the net result of his actions is to strengthen it immeasurably. As so often happens in Tolkien's work, "Out of evil cometh good".