Yeah about that. How on earth is Jon going to go claim a title he believes he's the rightful heir to, with just the North and the Vale? On top of that, there's a particular lore video that puts doubt as to whether Jon ever truly was first in line to the Targaryen crown. Namely in that Aerys intentionally skipped Rhaegar's line (bypassing his known legitimate children) when he died, and picked Viserys as his heir.
Also, and if I'm mistaken someone correct me, but when Jon joined the Night's Watch and Viserys had died, Dany became the Targaryen queen correct? She isn't just the heir to her family's crown, because just like Viserys was considered a king in exile, she too became queen. Meaning that Jon after his resurrection becomes her heir, and he doesn't unseat her from the title she has already achieved. Once you rightfully (as Visarys was dead and Jon was in the Watch) gain a position, succession starts after you and not rolled back. For example, if by royal decree Ser Jorah is given a pardon (which I think he got), he doesn't suddenly unseat Lyanna Mormont as ruling Lord/Lady of Bear Island. Until she has children of her own, he becomes her heir or falls behind any other named heirs.
Now before some think that getting the Iron Throne is what's needed to become queen or king, that's not true. A former ruling house in exile can still claim a throne, and have a position of King or Queen in exile, in much the same way that the Stuarts for example did when they lost the British crown. The Stuarats still refered to themsleves as Kings and had the same succession system as any royal house would, even though they weren't in Britain to rule. So Dany was for all intents and purposes already the Targaryen Queen before Jon got out of his Watch oath. I think he's her heir, due to Dany not being an heir but the actual Queen. Plus that's if we ignore what I mentioned earlier about what Aerys did.
