Ah, the royal house of Arnor. The first king of Arnor was Elendil, who was simultaneously high king of all faithful Numenorians in exile. His sons Isildur and Anarion were co-kings of Gondor. When Isildur rode north to Arnor after defeating Sauron, it was to take up the Scepter in Annuminas and rule as High King over both realms, with his nephew as king in Gondor. But since he never made it back to Arnor, things didn’t go as planned. after the fall of all 3 of those individuals, Valandil was being groomed to rule in Arnor, Meneldil was king in Gondor, and noone was claiming to be high king.
For many centuries, both kingdoms had their kings, and muddled along, with continued communication and acknowledgment of their shared heritage. Noone in Arnor tried to reassert their right to rule in Gondor until Arvedui (Last-King) of Arnor, when the throne of Gondor became vacant and there was no clear-cut heir.
Arvedui had argued that Isildur had never intended to give up the royal status of himself or his line in Gondor, but rather to be High King over Gondor and Arnor, as his father Elendil had been.
Arvedui also laid claim based on his marriage to Firiel, daughter of Ondoher, King of Gondor who’d just died in battle.
But the High Lords of Gondor weren’t buying it. They believed that the throne of Gondor belonged to the house of Anarion exclusively, and that furthermore, any of Firiel’s offspring would not be eligible due to the need to pass the kingship via the male line in the land of exile. Things had been done differently in Numenor, they admitted (there were several ruling queens in Numenor) but life was too uncertain in Middle-Earth to do such things there.
Arvedui was in no position to go against the wishes of the Dunedain of the south, so he didn’t press his claim. The throne of Gondor was given to Earnil, a direct male descendent of Anarion. His son Earnur was to be the last king of Gondor.
By the time Earnur died, the northern realm had been reduced terribly, and Arvedui’s descendents had too many other things to worry about (like surviving) to press a claim again, which had already been rejected.
Finally, as others have pointed out, Aragorn had to not only show up, but do such remarkable deeds that he would be remembered along with the likes of Beren, Earendil, and Elendil before he would be accepted High King of the Reunited Kingdom.