For people like Miyashita Naoyuki, with his narrow-mindedness, arrogance, and a twisted psyche due to his background, you either have to swallow your pride and avoid confrontation to maintain superficial amicability or destroy him completely so he cannot retaliate.
Aoyama Xiu Xin's relation with him had deteriorated, but his relations with Miyashita Hidekatsu, Miyashita Ritsuki, and Miyashita Ryouko were good. Thus, dealing with Naoyuki required considering their feelings; otherwise, offending the entire Miyashita family because of him would be more costly than beneficial.
Miyashita Ritsuki, who never liked Naoyuki, wasn't the issue. The focus was on Miyashita Hidekatsu. The plan was to let Hidekatsu gradually become disappointed and disillusioned with Naoyuki, steadily diminishing Naoyuki's value in his eyes, even to the point of making him loathe and fear Naoyuki before fatally dealing with him.