A little Bit of Meta and Some Predictions for "The Search"
This is all true, if this was where the writers were going, and it's a common enough reveal, but there are three reasons I'm not worried at all, and fandom, which is full of smart people who love to analyze things, is for some reason not talking about any of these things.
Firstly, if Zuko were not Ozai's son, he would lose his claim to the Fire Nation throne. We know that he rules until he abdicates late in life and passes it to his daughter. Now, it's conceivable he could hide his heritage, and destroy all records, and remind everybody not to listen to his sister, because she's crazy, but this is Zuko we're talking about, who not only hates to lie and has an overdeveloped sense of honor, but is also really bad at it. It would eat at him inside until it came out, and then he would have civil war. That would make it really difficult to accomplish all the cool things we know he accomplished as Firelord, wouldn't it?
Secondly, Bryke have openly said they modeled Ozai on Zuko and what he would look like as a grown man. The physical resemblance is extremely strong between father and son. You cannot tell me that this man:
is not the son of this man:
Heck, in "The Promise", when Zuko clearly wasn't eating right, and his cheeks were sunk in, the resemblance was even stronger. Young Iroh in Korra has an even stronger resemblance to both of them, being in the middle of their two ages, and we know he's Zuko's grandson. No, there is a strong genetic link between Zuko and Ozai.
Thirdly, it makes no sense to have the big reveal about Zuko's parentage so soon. If Zuko really were illegitimate, I would expect to find out in part two or three of "The Search", not part one. That says to me that it is set up.
What does make sense, however, is that Ozai thought Zuko wasn't his. It would certainly be a reason to show such blatant favoritism, and why he went from favoring Zuko just as much as Azula, as shown in "The Promise" to completely disregarding him so quickly. I bet he nearly had a heart attack when Zuko came home looking just exactly like him. I bet Ozai suddenly having time for him in season three had as much to do with that as with Azula telling him he brought down the Avatar.
It also sets up a further contrast between Ozai and Iroh. Ozai sees a loving, eager to please boy he believes is not his son, and mistreats him and casts him aside. Iroh sees a resentful, sullen, defiant teenage boy he knows is not his son (yes, he's his nephew, but I doubt their relationship is predicated on Zuko being Ozai's child, in fact it's probably a strike against it) and takes him in and cares for him, and does his best to raise him into becoming a good man.
And it gives Zuko an oppertunity for a little more character growth and some closure. Zuko is still desperate for a father. He has Iroh, but there is still a little boy, who desperately wants his dad to love him, and if that dad isn't Ozai, so much the better. It gives Zuko the chance to hope that Ozai isn't his father, before making him accept that yes, Ozai is his father, Ozai is a nasty horrible human being, and Zuko doesn't have to be. Funnily enough, that would address directly the cultural narrative that many fans are so afraid will play out in "The Search", and it would very much be in the style of Bryke and A:tLA thus far.
What I would love to see (though doubt I will) is for Ozai at the end of this to act like "Well, now you know why I treated you like garbage, and now that I know you're my real son, you should forgive me," and Zuko being all "No, you're a horrible person, rot in prison."