Going back to the alleged primacy of male heirs, the son of King Stephen was William, but his successor was Henry, the male heir of Matilda the female heir to the throne of England that Stephen usurped. Alfred the Great, by the way, succeeded his brother Aethelred, not either of Aethelred's sons. As near as I can make out the original British law that is being used is Henry VIII's decreed succession.
The supernatural elements of the show are there as flavor only. The core of the show is the individual character arcs.
Character is what you do, what you want. For fictional characters, the supernatural is directly implicated in what they can do, what they might fear. There is no separating the supernatural from character, any more than there is separating society from character, or the natural world, for that matter. It's just bad writing to pretend.
In the books Dany was conceived during the last month of Robert's Rebellion. She wasn't even born until after her mother fled to Dragonstone, nine months later. Obviously since the kidnapping of Lyanna Stark set off Robert's Rebellion, Jon was likely conceived earlier than this. According to the show (books haven't revealed yet) Ned finds him as a newborn fairly after the Battle of the Trident, though we don't know exactly how long. Given Rhaegar was afield for some time, Jon must be at least a few months older - possibly close to a year.
DigiFic Writer is right on this however. This timeline has one thing right. Rhaegar must have impregnated Lyanna before the Trident. The notion that Aerys must have impregnated his wife just before he was killed has nothing to support it. The notion that they murdered Rhaegar's heirs upon taking King's Landing shows they wouldn't have let Aerys' new heir (Viserys) and spare (Daenerys) unmolested for nine months. This is not just unsupported but absurd. The episode Stormborn talking about the storm when Daenerys was born was even compatible with the great storm, Daenerys' birth and the fall of King's Landing being close in time. And last, the idea that Ned Stark left the war immediately after the battle of the Trident and quickly found the new-born Jon is highly implausible. It's not even a sound interpretation of Ned's remarks about searching for Lyanna, as I remember.
And if Raenys/Aegon don't count as the marriage was anulled:
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In all cases Jon is ahead of Dany.
I think it is impossible for the annulment to be valid. There is no such thing as a secret annulment, because Ellaria the wife would have to be able to make her response in court, either to affirm or contest the grounds for annulment. The idea that a mere High Septon could annul a royal marriage on his own is preposterous. The King himself might be able to, although not certainly, but Rhaegar wasn't King. A plural marriage seems no more an obstacle to inheritance than an incestuous one, but no, it is not clear that Jon is the one true king though he does have a claim. It would have been vastly stronger if Rhaegar had acknowledged him as his son and heir, though. Varys should have known about Jon's impending birth, just as he knew about Robert's bastards. It is also likely Aemon knew, and Jeor Mormont (explaining their favoritism to Jon.) Qyburn should have found out about it for Cersei, who should be interested in holding Jon Snow's southern mother as leverage. Even making the claim Jon is the heir serves her purposes.
Aegon's claim to Westeros was dragons. The same should be true of the later members of the dynasty. Indeed, Aerys' obsession with fire is either Bran driving him mad like Hodor, or because Aerys knows without a dragon he's not the true king, a knowledge that unhinges him.