Livia was 'insane' because she was both an orphan and lost all those she cared for. We don't know the circumstances surrounding her early life or even how her parents came to be killed - but we do know that she's definitely a pure Garlean given her sister's revelation. Which makes it a possibility that she's an orphan as a direct result of Garlemald's enemies killing her parents - which would at least partially justify her ruthlessness and hatred for those who aren't aligned with Garlemald. That doesn't excuse her attitude/behaviour, naturally, but it's not as if there's a lack of Eorzeans who slaughter Garleans indiscriminately with no regard to the fact that they have hopes, dreams and loved ones.