Bran is literally called Robb's heir in both the show and the books though. Yes it's only because Robb has no children but as the second son of Ned Stark, Bran was second in line and therefore when Ned died, Bran became Robb's heir. In the books he literally only considers legitimising Jon because he believes Bran and Rickon, the next two heirs, to be dead
Bran is named the heir to Winterfell, he is not at any point named heir to the Northern Monarchy. The Northern Throne and Winterfell are not the same thing, they are separate entities. Bran IS second in line to his father’s position, he is not automatically second in line to his brother’s position, and Ned was never KitN.
I think you are trying to argue here that the North has a hereditary monarchy? But succession in hereditary monarchies doesn’t move horizontally, it moves vertically. Bran is his father’s heir apparent, he is not his brother’s heir apparent. If Robb had specifically and explicitly declared Bran the heir to the Northern Throne, or if he hadn’t named an heir at all, there might be a better argument for Bran being the presumptive heir (presumptive heir: a person entitled to inherit a throne, peerage, or other hereditary honor, but whose position can be displaced by the birth of an heir apparent, or by a new heir presumptive with a competing claim to the position in question). But even a presumptive heir can still be beaten out or unseated by an heir apparent or a presumptive heir with a valid claim (see Jon Snow).
The World of Ice and Fire companion app, written in conjunction with GRRM’s input, confirmed that Robb named Jon as his heir and legitimized him at some point. Idk exactly where the link is, I will have to go digging for it. BUT, in the books there is a conversation between Robb and Catelyn about declaring Jon as his heir. He then makes all his lords sign a paper about declaring his heir, but they don’t tell us who Robb declares as his heir.
In A Storm of Swords, Robb and Catelyn have the following conversation:
“Mother.” There was a sharpness in Robb’s tone. “You forget. My father had four sons.”
She had not forgotten; she had not wanted to look at it, yet there it was. “A Snow is not a Stark.”
“Jon’s more a Stark than some lordlings from the Vale who have never so much as set eyes on Winterfell.”
“Jon is a brother of the Night’s Watch, sworn to take no wife and hold no lands. Those who take the black serve for life.”
“So do the knights of the Kingsguard. That did not stop the Lannisters from stripping the white cloaks from Ser Barristan Selmy and Ser Boros Blount when they had no more use for them. If I send the Watch a hundred men in Jon’s place, I’ll wager they find some way to release him from his vows.”
He is set on this. Catelyn knew how stubborn her son could be. “A bastard cannot inherit.”
“Not unless he’s legitimized by a royal decree,” said Robb. “There is more precedent for that than for releasing a Sworn Brother from his oath.”
Robb then has a council with his lords as follows:
He picked up a sheet of parchment. “One more matter. Lord Balon has left chaos in his wake, we hope. I would not do the same. Yet I have no son as yet, my brothers Bran and Rickon are dead, and my sister is wed to a Lannister. I’ve thought long and hard about who might follow me. I command you now as my true and loyal lords to fix your seals to this document as witnesses to my decision.”
In the show Robb never actually named anyone heir to his throne, presumably because Talisa is pregnant with what they believe is a son. At that point he probably didn’t think he needed any backup or contingency plan.
Personally I would quibble over the argument that Robb would “only” have legitimized Jon because he thought Bran and Rickon were dead. Jon and Robb were the closest of the Stark siblings only (arguably) after Jon and Arya, but that’s kind of beside the point tbh. On the show everyone does think that Bran is dead, therefore there is NO heir (presumptive or apparent) left for the Northern throne (you could make an argument for Sansa being presumptive heir, but as we know Westeros is patriarchal af). So basically, after the battle for Winterfell, the Northern Lords named Jon the elective monarch (the same thing they did when they declared Robb KitN).
I am not saying that Bran has zero claim or rights to the Northern Throne, I am just saying that it’s not conclusively his the way it would have been had he been Robb Stark’s son. It’s not that the title of KitN can’t be his, it’s just that it isn’t automatically and unquestionably his; it’s not as pure and simple black and white as you’re making it out to be.