Okay so if the Angels are really her people, why the desire for change?
Because she wants to turn the angels into something better then they are, and she wants them to be accepted into the galactic community, which is something that being slavers directly interferes with. Also, keep in mind that while the angels are her people, she still thinks of herself as Matari, and arguing about what it means to be matari is rather what started her down this path in the first place.
I've always maintained (I'm late to the Angel party, but I make up for it with chutzpah) that slavery within the Cartel is a matter of economics. The Cartel engages in slavery largely to sell slaves (and I'm not talking about the underclasses of Cartel society, I'm talking about actual "we have captured you, we are now going to sell you into slavery" slavery), and they do this because their
customers want to buy them. If the market for slaves dried up, they'd shift that aspect of their business to black market auto parts, or long limb roe, or bobbly-headed SOE nun dolls, or whatever people want to buy.
So maybe Nikita has a vision of a Cartel that sits equal in status to the other big empires, and she's decided that it's practice of slavery is standing in the way. She could decide to focus on eliminating the
market for slaves, figuring that by doing this she'd coax the Cartel away from this practice. There'd surely be opposition within the Cartel - she's working to eliminate a lucrative market for them. At the same time, she could bring other Cartel members to her view by arguing that slavery within the Cartel is a matter of a focus on short term profits blocking a more profitable long term strategy. Likewise, she might forge some ties with, say, the Republic by bringing her organization into the fight against Amarrian slavery.