Queens don't really sit the throne in Westeros "nowadays" (nor in the days before Aegon the Conqueror).
If Dany takes the Iron Throne, only folks over a certain age are likely to be comfy with her leading rather than say Aegon. In the days the Targaryens wore the crowns, this (and their incestuous ways) were one of a few Targaryen practices that were mostly tolerated in Westeros because they had the dragons and power (later just the power). Perhaps the smarter Westerosi saw the horrendous mess that Cristos "the Kingmaker" Cole left behind when he nudged Aegon II towards making himself king when his elder sister had been chosen by their late father to succeed him.
I tend to see Aegon as the Mummer's Dragon. He's Illyrio and Varys' Blackfyre pretender. Varys made the attempt on Dany's life to clear the way for Aegon.
Aegon's immediate purpose in the story is to put another side on the field (now that at least one has been taken out). The aim for Varys/Illyrio is for him to get the Martells on his side if possible and take on the considerably diminished forces that remain in Westeros. A Blackthorne on the Iron Throne with John Connington's help.
However, that's not likely the story Martin winds up telling. He's already foreshadowed that the Mummer's Dragon is a threat to Dany, or at least someone to look out for. My take on it is that he's not one of the three dragons. They'd be those two secret Targaryen bastards, John and Tyrion. My thinking is that Aegon and friends get pretty far, but it all falls apart somehow. Possibly even Dany herself does him in? Or one of the dragons, when he tries to take a ride.
The thing to realize with all the focus on legalities is that birthright or no, the leader with the most swords/ships who uses them most effectively is more likely to gain and maintain power.
A birthright can certainly bring swords to his banner, but we need to remember that Viserys had a better claim to the Iron Throne than anyone. Was anyone in Westeros (past Doran) even dreaming of trying to put him on the Iron Throne?
As far as anyone who's not in the cave with Bran knows, Bran is dead. That could get a bit complicated when/if Rickon/Osha appear, but I doubt it matters much unless he actually makes an appearance. Which he's rather unlikely to. Rickon is useful to Manderly in that he can be used to help supplant the Boltons while easing the minds of other northern lords over his apparent betrayal of the Starks in favor of Joffrey/the Freys. If Manderly unites the North, that leaves him to run things until Rickon matures. Which makes for a fair bit of time to feather his own nest...