Again, what does "king of the north" mean?
It's an honorary title to the person who leads the independent kingdom of the north.
They're not rebelling against the crown.
They are united against the whitewalkers, and Jon is the right man for the job. Not Bran. Not Sansa. Certainly not Littlefinger. Jon.
Robb Stark once said "Bran can't be lord of Winterfell before me." He has no contact with Jon Snow after that scene, so why would he suddenly choose Jon in the line of succession. Show Robb is still holding out hope that Bran and Rickon made it out, as is Catelyn. That's my interpretation anyway.
This season, the northerners have been stubborn in their refusal to support Jon. They call him a bastard. They say he's a deserter. All the problems you are describing were addressed. Now, at this time after Jon led the attack against the Boltons, they finally pay him the honor of a title that just amounts to "leader", which is what he is.
They are rebelling against the crown. That's what LF means when he says the Vale has declared for the North.
Like do you think Cersei is okay with the Starks taking over the Norh again?
In fact calling him the King in the North is acknowledging that the highest power in the north is Jon not whoever sits the iron throne