I think the experience of being an underdog has changed Azula's perspective in this universe, just like Zuko must be a different individual in this reality due to being the favorite. I've found that I was a lot more empathetic and less out-of-touch after I suffered a few setbacks in life, and that's probably the same for Azula even if she won't be crowned Queen of Empathy in any universe. Tyzula is a fascinating pairing, even though I can't see a plausible way for it to work in the canon universe--but there's always AU for that. :)