This is such a good post and a juicy topic!
I also feel like CDs are extremely traditional and patriarchal, with a side dish of “fit into the structure, and then you can do whatever you want”. And isn’t it ironic, that even among the Celestial Dragons, the role of the oldest sibling never changes? Doffy takes responsibility for Rosi the moment he realizes that his parents cannot protect or feed them. He even takes Rosi to gather food for their mom, completely taking over the role of the “man of the house”. (Their mom was sick, so no shade there, but Homing was never depicted doing anything active to help them…)
I fully agree with your analysis on the expressions of love: Doflamingo is extremely indulgent with his family but never tries to “fix” them, neither when it’s real mental issues that require help (like with Baby 5), nor when it’s things people cannot change and thus shouldn’t be bullied for (Pica’s voice). Plus, he learned from his Executives that when you defend family, you go all out, and destroying a town for a slight offense is how family acts, actually.
I think the DQ parents were definitely your typical noble parents who mostly left child-rearing to others. The fact that Homing never taught them that slavery is wrong before moving to the North Blue proves it. Although I do think DQ mama was closer to her children than your typical noble lady, judging by how Rosi clings to her and how attached Doffy was to her. I personally like to think that she spent time with them and built strong emotional connections with them, but only the “fun” parts. All the unpleasant things like tantrums and diapers were left to the servants.
And neither of the DQ parents were able to set any boundaries or discipline the children, this much is clear. Judging by how soft and passive the parents were depicted, I think they were probably completely overwhelmed by Doffy’s energy and aggression. He was the loud one, the outgoing one, the one with the “fight” response to stress. He was not the easiest child to deal with, but he also never had a parent figure who would set boundaries for him or offer him guidance and protection - which is extremely exhausting and frustrating to a small child.