<p>Zuko was learning from Iroh for what? 2 years? Still didn't master the older ways. You don't need to meet the dragons to learn about true firebending, seeing as JJ had the same vision as Iroh and there's nothing that suppose he met them(i have a theory, though). Zhao was less skilled than Zuko even if he was learning for the sake of power and destruction like you say, and he was very adept of this way of firebending. Zuko was impatient and incapable of mastering the basics even after training for 2 years under Iroh that was teaching him the true ways of firebending. Azula mastered more forms than Zuko when they were kids even though she is younger, because she trained more while Zuko played with his mother and possibly didn't train as much. Azula was shown as someone who wouldn't stop training until things were perfect, even a hair out of place calls for a repetition. When he tried to make a form for his grandfather he didn't fail at firebending, he tripped on his own feet. Yeah, Azula practiced, practiced, and practiced, while he never mastered basics by learning true firebending from Iroh.
</p><p>After meeting the dragons he said: "I lost my drive so i couldn't firebend, but now <i>i found a new drive</i>, help you stop my father." or something like that. It doesn't matter if he had time to relearn because the dragons didn't teach him anything, they showed what firebending really is. He already knew everything he needed because Iroh had already taught him about fire coming from the breath and stuff. All the moves, forms, etc. will be the same for the two types of firebending, so he didn't have to relearn anything. Iroh was a strong firebender before meeting the dragons and he wasn't a power-hungry like Azula or Ozai or else they wouldn't find him worthy, so even if he's not malicious like Azula, he could be a more skilled firebender. Zuko eing weaker can be explained by one of the two following points:
</p><p>1 - He didn't train as much as Azula.
</p><p>or
</p><p>B) He's not a firebending genius like the rest of his family.
</p><p>I actually prefer the first option.
</p>