Why would someone assume she had any illicit relations with Aozora? They interacted, they went their separate ways. She hasn't even done anything with Enishi, either. Or anyone else in the family. Just Nasa. What a weird conclusion to make. Pretty sure she's never banged anyone in history before, and it's even still up in the air if she's done it with Nasa, even, considering it's been shown Toast interferes with their attempts in the past, along with various other interferences in other cases, and they still act like they haven't seen each other's privates.
Anyway, as a reminder... Shotoku's line ended, if you thought Nasa was related to him. They literally mentioned the fact all his descendants died out in the manga, even. According to Wikipedia, I see all of his children listed as dying on the 30th of December, 643. Even the daughters. Someone really had it out for his bloodline. Looking into it, they all committed suicide after being attacked by the son of a court rival.
Ergo, Nasa cannot be related to Shotoku. In fact, nobody can be directly related to him. Again, the bloodline was destroyed. The Yuzaki family is unrelated. Same for the other historical Japanese figures she's been shown with in more recent chapters: they're most likely unrelated to the Yuzaki's, pretty sure, and they're notable figures of history you can actually look up about.
Michizane,
Masakado,
Ieyasu...
But, damn, yeah. She basically saved the Yuzaki family from ending up like Shotoku's line. And saved her own future in the process.