By the time of Aragorn, the possible heirs to the throne had dwindled. As I recall I thought there was as an explanation that some had married people not of Númenórean descent and so were disqualified.
However, I thought there was at least one King of Gondor married to a woman not of Númenórean descent (leading to the Kin Strife), and also Arwen is technically not Númenórean. Does marriage outside of the Númenórean bloodline disqualify a person for being the monarch of Gondor?
As far as I can tell, the evidence (from the LotR Appendices) is:
Rómendacil showed especial favour to Vidugavia, who had aided him in the war. He called himself King of Rhovanion, and was indeed the most powerful of the Northern princes, though his own realm lay between Greenwood and the River Celduin.29 In 1250 Rómendacil sent his son Valacar as an ambassador to dwell for a while with Vidugavia and make himself acquainted with the language, manners, and policies of the Northmen. But Valacar far exceeded his father's designs. He grew to love the Northern lands and people, and he married Vidumavi, daughter of Vidugavia. It was some years before he returned. From this marriage came later the war of the Kin-strife.
'For the high men of Gondor already looked askance at the Northmen among them; and it was a thing unheard of before that the heir to the crown, or any son of the King, should wed one of lesser and alien race. There was already rebellion in the southern provinces when King Valacar grew old. His queen had been a fair and noble lady, but short-lived according to the fate of lesser Men, and the Dúnedain feared that her descendants would prove the same and fall from the majesty of the Kings of Men. Also they were unwilling to accept as lord her son, who though he was now called Eldacar, had been born in an alien country and was named in his youth Vinitharya, a name of his mother's people.
It seems clear that until that time, marrying out of the Dúnedain disqualified the progeny from the throne and it had not previously been an issue and so, presumably, had not been done.
Arwen was not a Númenórean, but her blood was higher than that of any Númenórean — she was the niece of Elros, the first King of Númenor after all! and Noldorian on her mother's side. Even if someone had wanted to try to exclude her descendants, they would be very unlikely to succeed.
