Sorry, Exa, but Tolkien is very clear on this. There should not have been any seperate Kings of Arnor and Gondor. The true Kings of both Realms are the ones descended from Valandil. The line that took the throne of Gondor were NOT technically legitimate.
They became legitimate simply by time and tradition, but to men who swore an oath to Isildur, they would NOT consider that line in any way relevant- onlky the Arnorian line.
Regardless of the bloodlines of those who called themselves King, the true Kings of Gondor for the ENTIRE Third Age were Isildur's descendants in Arnor. The joint crown was theirs.
Isildur was not going North to take up the rule of Arnor. He was returning north because that was where the seat of power was for the ruler of BOTH Kingdoms- but, to quote Tolkien, "Elendil... was accounted the High King of ALL the lands of the Dunadain. While Elendil still lived, the conjoint rule to the South was committed to his sons, but when Elendil fell, Isildur departed to take up the High Kingship of his father, and committed the rule in the South in like manner to his brother. HE DID NOT RELINQUISH HIS ROYALTY IN GONDOR, nor intend that the realm of Elendil should be divided forever."
So you see, there NEVER should have been a seperate King of Gondor, only someone the King of Gondor AND Arnor- that being Isildur- left behind to look after things. When Isildur died, the person he left in Gondor ASSUMED the title of King. That assumption was NOT literally legitimate.
And so hence, that entire line would be disregarded by the Men of the Mountains.
So I am afraid your objections are incorrect, Exa. I said Earnur was not literally a King, and indeed, literally he was not, he was part of a line that erroneously assumed Kingship.
Isildur was NOT King over Gondor with his father, he, along with his brother, ran Gondor FOR the King. Gondor did not have its own King.
He did not crown Meneldil King. He left Meneldil in Gondor to run things for him. Isildur himself, as clearly stated by Tolkien, was still Gondor's King. The line descends through him.
They are listed as King in the chronology for simplicity's sake and because they called themselves Kings but the situation above is exactly as Tolkien describes it- this being an error, and the true line being in Arnor, all the way down to Aragorn.