On Ozai not being a well-written villain because of the lack of sympathy the audience has for him, I think that having Ozai be a sympathetic villain with pure intentions would generally be in bad taste, considering the role of the character is to build tension in the story by threatening a genocide. I think most people do not care for sympathizing with a character like that, much like with real-life historical figures who have committed the same deplorable acts. But even if most people did, I think that there exists a sympathetic reading of Ozai's character that is well-supported by the canon.
One can argue that Ozai's villainy is not so much a product of nature as it is nurture. Would Ozai have turned out the way he did if he had a family like Sokka and Katara's? If he had a father figure like Iroh? His character and actions line-up perfectly with what was expected of Fire Nation royalty, at the very least post-Sozin. Even Iroh was caught up in the Fire Nation's genocidal agenda until he went through the very traumatic experience of loosing his son; Ozai never went through such an experience, but what if he did? We assume it would not make a difference because Ozai just does not care and never cared about his children, but about Ozai being an irredeemably---if not inherently---bad father, here is something to think about:
"When he planned to cast his infant son from the palace, Ozai gave in after Ursa and the Fire Sages pleaded with him to give Zuko a chance. Ozai believed that he attempted to be a good father in his early years, a view shared by both Zuko and Azula. However, his pride, anger, and lust for power degraded his relationship with his family."
You can argue that he was ruthless, ambitious, proud and selfish because he was born and raised in an environment that encouraged and celebrated those very traits. The existing canon also implies that he was just born like that, which would also support a reading of Ozai as maybe struggling with a mental disorder, and I think some people can empathize with that.
I think that Ozai served his purpose in the story. As per the design of the creators, he was supposed to be this invisible, yet not entirely enigmatic, evil force plotting in the shadows, waiting for the right moment to emerge. All the other antagonists, like Zuko and Azula, acted as Ozai's extended organs and served their own purpose in the story. Part of those purposes required them to have tragic back stories and complex character arcs that rendered them more sympathetic. In my opinion, Ozai's role in the story did not hinge on either of those because he was designed to be a figure that Aang and the other characters feared or despised, not a lost soul to be understood and redeemed. In fact, giving Ozai something like a tragic back story might have undermined that purpose. At the end of the day, some people will vibe with that, and others won't. However, I don't think any one person's opinion and taste should become the standard, which is why I would avoid language such as "a villain is supposed to..." I don't think a villain is supposed to be anything other than what the story needs them to be.