In the search, we see flashbacks to how Ursa and Ozai got married.
Azulon says that the sages foretold that if the descendent of an avatar married into the royal bloodline, they would spawn a line of very powerful rulers (or something along those lines).
Why would Azulon want Ursa to marry Ozai? At the time, Iroh was in line to be firelord. Ozai's children were not supposed to ever be rulers of the fire nation, at least not according to Azulon's plan. He would have wanted her to marry Iroh or even Lu Ten (idk how old he was) . Perhaps Iroh was already married or whatever but, did Azulon really think "Hey, if something happens to Iroh THEN the firenation will be ruled by this powerful line"? I don't think so, he seamed to favor Iroh over Ozai. He was outraged when ozai proposed he be firelord.
This whole plot point is so poorly thought out and it shows why 1. the original writers should write everything and 2. All stories should have a "writer" who acts as an observer who proof checks and tweaks things. Zuko being the descendent of Sozin and Roku should have been more of a "it just happened" thing. That would have made more sense than this poorly thought out backstory.
The more I think about it, the more I think my personal cannon is just the original series. No comics, no LOK.