It's hard to say; when she said the divine right to rule is something you're born with, I'm not sure if she meant that your overall status has anything to do with it. Long Feng was conniving, but either the fact that he was born with nothing played a part in his downfall, or Azula was just more intelligent then him.
I'm going to do with the second one. Azula had planned out Long Feng's moves. She knew, ang hoped, that she would be take to Long Feng (or whoever was in control of the Dai Li) She is a very good actress; even the viewer would forget that she was hoping for this. And when Long Feng was betraying her; she didn't even bat an eye. She saw him as predictable, and in her eyes, predictable does not work in situations like this.
If Azula had been born like Long Feng, but still had her intelligent, calciulating personality, she probably would have been able to do more that "claw her way" to her goal.
The way I see it; Azula's mind works like both a chessplayer, and a roadmap. With both, you find different directions you take in case there is an obstacle. You plan on something getting in your way, but don't be surprised by it, find a way around it. She is always several steps ahead of someone and she is rarely surprised.
But, once and a while, she even slips. What happened with her, Mai and Ty Lee, that was probably the one time something completely off guard. And it will play a big part in her mental deterioration. It's kind of funny that after 14 years of being unnaturally intelligent and physically gifted, being able to get what she wants through manipulation and downright charisma, this one event will be the fray in the thread that will start the unraveling.
..That's weird; I'm more insightful when I get very little sleep. :O