The Heirs of Elendil had only one more chance to restore their Kingdoms
Arvedui of Arthedain claimed the Throne of Gondor after the death of Ondoher in TA 1944. Still young at the age of 80, he claimed the throne via two routes:
In his own right as direct descendant of Isildur, King of Gondor
On behalf of his wife, Fíriel, the daughter and only surviving child of the late King Ondoher
But strongly influenced by the Steward Pelendur, the Council of Gondor dismissed both of these claims:
The crown and royalty of Gondor belongs solely to the heirs of Meneldil, son of Anárion, to whom Isildur relinquished the realm. In Gondor this heritage is reckoned through the sons only, and we have not head that the law is otherwise in Arnor.
--- "Gondor and the heirs of Anárion" (The Lord of the Rings, Appendix A, Part IV)
The Crown was instead awarded to the victorious captain of Gondor's Southern Army, Eärnil, the great-great-grandson of Telumehtar Umbardacil (reigned 1798-1850.) Eärnil had defeated the enemy army that had invaded Ithilien from Harad and then headed north, gathered the remnants of the routed Northern Army, and destroyed the eastern invaders, the Wainriders.
A pedigree is less important to the people of Gondor than military success
The memory of the Dúnedain was long. Just as the Heirs of Isildur never forgot their ancient majesty as the High-Kings of Arnor, the Council of Gondor lived in fear of repeating the errors that led to the Kin-strife, Gondor's civil war, which was fought between the factions led by:
King Eldacar, the half-Gondorian (but still legitimate) son of King Valacar.
Castamir the Usurper; Captain of the Ships, and although he was the great-grandson of King Calmacil, one of the closest collateral heirs to the throne.
Valacar was the first Prince of Gondor to wed anyone other than a Númenórean, and rebellion was beginning to stir by the end of his reign. The war was brutal, and it lasted for some time until Osgiliath was taken. Eldacar fled to the northlands and his mother's people, but his heir, Ornendil, was among the many who were slaughtered and the Master-Stone of the South, the Palantír of Osgiliath was lost.
Castamir had not long sat upon the throne before he proved himself haughty and ungenerous. He was a cruel man [...] love for Castamir was further lessened it became seen that he cared little for the land, and thought only of the fleets and purposed to move the king's seat to Pelagir. [...] when Eldacar, seeing his time, came with a great army out of the north, and folk flocked to him
--- "Gondor and the heirs of Anárion" (The Lord of the Rings, Appendix A, Part IV)
So, after ten years of brutal rule by Castamir, Eldacar took his chance and was able to recapture the throne. He killed Castamir, but his sons and many of his supporters escaped to Umbar, which remained a dangerous enemy of Gondor until it was finally recaptured by Aragorn II.
Paradoxically, thanks to the Kin-strife killing off so many of the "best and brightest" of Gondor, the Kin-strife itself led to the very thing it had been fought over. The Kings kept their close kin under suspicion, and those same kin would either flee to Umbar or renounce their status and take wives who were not of Númenórean blood. So by the time of Eärnil's death, there was no acceptable claimant to the throne.
The people of Gondor had some very strong ideas about who should be King
The prophecy that gave Arvedui his name foretold that the Dúnedain would face long suffering, but would rise again before they would be reunited.
"Arvedui you shall call him, for he will be the last in Arthedain. Though a choice will come to the Dúnedain, 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 Dúnedain arise and are united again."
--- "Gondor and the heirs of Anárion" (The Lord of the Rings, Appendix A, Part IV)
Prophecies in this world have been shown to be serious business. But between the Fall of Arthedain and the Return of the King, there were no substantial changes to the fortunes of the Dúnedain.
They know that prophecies are real, and should not be ignored
When the Council of Gondor denied Arvedui's claim, he made this response:
"Elendil had two sons, of whom Isildur was the elder and the heir of his father. We have heard that the name of Elendil stands to this day at the head of the line of the Kings of Gondor, since he was accounted the high kings of the Dúnedain. While Elendil still lived, the conjoint rule in the South in like manner to the son of his brother. He did not relinquish his royalty in Gondor, nor intend that the realm of Elendil should be divided forever.
"Moreover, in Númenor of old, the sceptre descended to the eldest child of the King, whether man or woman. It is true that the law has not been observed in the lands of exile ever troubled by war; but such was the law of our people, to which we now refer, seeing that the sibs if Ondoher died childless."
--- "Gondor and the heirs of Anárion" (The Lord of the Rings, Appendix A, Part IV)
The Council of Gondor made no response to this, but went ahead and gave Eärnil the crown.
The years grew longer, but the fortunes of the Chieftains of the Dúnedain had only slipped further. No longer the Kings of Arthedain, they were now forgotten by most and now led their people from the shadows.
"How many hundreds of years needs it to make a steward a king, if the king returns not?" [Boromir] asked. "Few years, maybe, in other places of less royalty," [Denethor] answered. "In Gondor ten thousand years would not suffice."
--- "The Window on the West" (The Lord of the Rings, Book IV, Chapter 5)
This is the kind of attitude that the Heirs of Isildur would have faced had they made another attempt to claim the throne. Would a people who would wait ten thousand years for the return of their king accept a "such a [king], last of a house long bereft of lordship and dignity," to use the words of Denethor?
"Men of Gondor, hear now the Steward of this Realm! Behold! One has come to claim the kingship again at last. Here is Aragorn, Son of Arathorn, Chieftain of the Dúnedain of Arnor, Captain of the Host of the West, bearer of the Sword Reforged, victorious in battle, whose hands bring healing, the Elfstone, Elessar, of the line of Valandil, Isildur's son, Elendil's son of Númenor. Shall he be king and enter into the City and dwell there?"
--- "The Steward and the King" (The Lord of the Rings, Book VI, Chapter 5)
Aragorn II was the first Chieftain of the Dúnedain to have the achieved such success. He defeated the forces of Mordor, led the victorious armies of Gondor and her allies, and reforged the sword of Elendil. The Line of Anárion had come to its end hundreds of years ago, but with all his achievements, and the ol' "I'm actually the Heir of Elendil instead of the Heir of Isildur" dodge,
Aragorn II was the first since Arvedui to have any chance to reclaim the Crown