It just occurred to me that it's pretty obvious Rhaenyra's side wins this. She's the only one who knows the prophecy at this point. Since we know Rhaegar knew the prophecy as well, you can easily conclude he descended from her.
Book readers still debate who actually "won", and usually conclude any victory was pyrrhic. I believe this is the point GRRM wanted to make.
Yes the Targaryen line continues through Rhaenyra and Daemon, so they "won" the war of bloodlines.
But Aegon II won the war of legitimacy, as he goes down in the history books as the rightful King, and Rhaenyra goes down in the history books as a usurper that attempted to steal her brother's birthright.
Rhaenyra is not listed in the history as one of the rulers of of the Seven Kingdoms despite the fact that she sat on the Iron Throne and ruled Kings Landing for a time. Her short reign on the Iron Throne is considered an illegal occupation in the official records. Also Aegon III is considered Aegon II's heir, instead of his own mother.
They say history is written by the victors, and the history and records says Aegon II won the war of legitimacy over his traitor sister and put an end to her tyranny.
By the time of Game of Thrones, Stannis considers Rhaenyra a usurper that died a traitors death.
The faith says the gods judged Rhaenyra as unfit, the smallfolk remembers her as Maegor with teats, in her own family Targaryen history lore she is remembered as the woman that caused the city to kill off the dragons, and the law says she was a pretender and rebel. This is her victory.
Aegon III and Viserys II both had the opportunity to honor their mother by legitimizing her in the public record, making her Rhaenyra Targaryen I, First of her name, but they never did. In the end, she only succeeded in being the mother of Kings.
