(I know this is probably contradicted by comics canon, but it will remain my headcanon until such time as I read the comics and incorporate them into my canon)
"Lucky to be born" implies to me that there were problems at Zuko's birth. Maybe he came early, maybe he was just sick.
So I've decided they thought he might die. And to Ozai, with his might-makes-right worldview, his sickliness meant he was an undeserving heir.
Zuko didn't die, and instead grew up to be a healthy child. But Ozai's first impression of him was of was a baby that, in Ozai's eyes, was weak. That shadowed how he saw the boy for the rest of his life.
Azula, of course, had no such problems at birth.