Casually referring to the Silent Wind by her name, and advising that anyone become more like her is hilariously OOC for just about any Demon, let alone an Angyalkae.
Primordials terrify First and Second Circles, Adorjan especially. I do not see Azzaria being at all comfortable with Kiritsugu learning Adorjani Charms, nor would she suggest he partake of her philosophy. Azzaria is a Citizen, that means she's very competent for a First Circle, part of that competence is how to read and predict others, and use them to her advantage. A Kiritsugu who starts removing attachments becomes very hard to predict, and very unsettling to watch (Silence is Death after all).
I mean, who's to say he couldn't learn how to fleanse away Azzaria?