(Even though Ozai favoring Azula simply because she's more talented and reminds him more of himself is also a perfectly good explanation.)

I think if anything, Zuko reminds Ozai more of himself, and I think he hates him for it. Underneath all of his bluster and arrogance and megalomania, he's still also the little boy who couldn't please his father, and whose big brother was always better. Ozai wants to, and does, see himself in the talented Azula, but he also sees himself in Zuko, and it's the things he hates about himself, that he perceives as weakness that he sees. Of course, he also sees a whole bunch of other "weaknesses", like Zuko being a decent human being, but I think a lot of Ozai's cruelty comes from him projecting all the things he hates about himself and make him feel inadequate onto Zuko.

Zuko having some time to explore the idea that he's not really Ozai's son before being forced to come to terms with the fact that yes, he is, but he's his own person, not his father's son

It als just seems like a really Avatarverse Brykish thing to do.