She's done as much or more than Taylor has to end the civil war and provide stability to the aftermath. She's definitely got a better claim to be the legitimate ruler.
Let's assume for a moment that this was done completely without self-interest (it was not, she did it to keep her own reign safe, but let's pretend), just to make the Principate more stable and stronger regardless of whether it'd support her afterwards or not; and I'll give to you, with absolutely no caveats, she is indeed the legitimate ruler, she had the best claim and she ensured it was respected. Granted.
How exactly does any of that goes against "the nobles are a superior tier of being to everybody else"? She's the noble, the best of them in her own eyes, and she's in charge, dictating everybody else what to do. For good reasons, sure, let's say that. It's still her in charge and everybody else under. None of that in any way challenges the worldview of noble superiority that she believes in, in fact it confirms it - the superior Prince, the one who is
most noble among them, won and was put in charge.
She is the legitimate ruler of a nation that espouses noble supremacy. These two things are not in conflict. She wants to strengthen a nation that espouses noble supremacy. These two things are not in conflict either. The parts of Taylor's proposal that furthered those goals, she liked.
The parts that did not, which posited that maybe the peasants should be taught and given the chance to rise, that the organized religion should have a say in things, and the divinely ordained champions should be incorporated into the laws of the land to remove their extrajudicial status? All of those are admissions that the Princes do not stand uncontested above all, and she fought or planned to subvert them from the beginning. That is what makes me say she's unwillingly to limit the power of the nobility at an instinctual level - because she believes that the only ones who can dictate conditions to Princes are other Princes, nobody else. All of Cordelia's actions, when we examine them, are in accordance with that belief, because they
stem from that belief. That's all there is to it, in my view.
As an aside: Taylor certainly never expressed any desire to rule, no matter what the High Assembly said. Her having judicial power, which means, in modern terms, separating the judicial from the legislative, as any functioning state should, in no way is her "being put in charge". The head of the executive is the one in charge, and the First Prince is still head of both the legislative and the executive in Procer; the characters in the story might well believe that splitting the power of head of the judiciary from the title of First Prince and handing it over to Taylor makes her the ruler, but we, the readers, know that that's just dictatorial nonsense.
It's sad that she wasn't able to make that argument in character, but that is what she was angling for - not rulership.