It was not just a question of right and lineage but of politics, of being perceived as having the "stature" of a king of Gondor, an image thing.
Denethor (to Gandalf) on his pyre: "So! With the left hand thou wouldst use me for a little while as a shield against Mordor, and with the right bring up this Ranger of the North to supplant me...I am Steward of the House of Anarion. I will not step down to be the dotard chamberlain of an upstart. Even were his claim proved to me, still he comes but of the line of Isildur. I will not bow to such a one, last of a ragged house long bereft of lordship and dignity."
Aragorn would not have the necessary stature until he had done something amazing, delivering the city from the Enemy through the captured Haradrim fleet wasn't enough for Denethor (whose pride had swollen a long established policy, not invented it).
Consider that on the death of Ondoher (1944), who perished along with his sons in battle with the Wainriders, Arvedui of Arthedain put forth a claim to the throne of Gondor. He had two arguments: not only was he descendant of Isildur, the senior line, but also had wed Ondoher's daughter, Firiel.
His claim was rejected, for reasons that echo Denethor's words from the pyre. Led by the Steward, Pelendur, they argued that Isildur had relinquished rule to Anarion's son, Meneldil, and further that in Gondor, the sceptre never passed through the female line. Arvedui had further arguements, but they were ignored.
The annals commnet: 'Arvedui did not press his claim; for he had neither the power nor the will to oppose the choice of the Dunedain of Gondor; yet the claim was never forgotten by his descendants...'
The crown instead went to Earnil, who was descended from Telumehtar Umbardacil (r. 1798-1850), 4 king back. Earnil's son, Earnur, was the last king, but, by that time, Angmar had defeated and destroyed Arthedain.
But, to some, this was seen as a missed chance. Malbeth the Seer to Araphant, Arvedui's sire: "A choice will come to the Dunedain, and if they take the one that seems less hopeful, then your son will change his name and become king of a great realm. If not, then much sorrow and many lives of men shall pass, until the Dunedain arise and are united again."
I dion't know about this. Sauron's strategy was to slowly wear down the overt power of his foes: had Arvedui come to rule a reunited Gondor-Arnor (in truth, Arnor would have been a weak, depopulated appendage to an essentially Gondorian state.) Sauron's allies would have worked to weaken this overextended state. There'd be no ignored reserve of Rangers from which a revival could come, would be on the spot and able to defend the even weaker and more humble people from whom salvation would ultimately come.
With Calenardhon-Dunland the essential link between the realms, even in its under populated state, would a king decide to alienate this bridgeland, even if it promised to bring into the neighborhood a promising ally?
Maybe things worked out for the best.