Re: One more thing... And that's what makes Babs more the active parent, and why it would make more sense for Babs to offer to adopt Cass, rather than Bruce.
Right. And you know, the thing about the title "Bat family" is it usually mean two things--there's the extended Bat-family which includes Bruce, Alfred, Dick, Tim, Jason, Cass, Babs, Steph, Helena, etc. All the heroes in Gotham. Then there's this nuclear family of Bruce, his dad, Alfred, and his sons.
And by having Bruce adopt Babs it not only makes it seem like of course Bruce is the only one with the authority to adopt children etc., but it keeps the distaff side of the family from being a family. It's like their relationships have to be more unstable (Babs can just break up the team and then they're all strangers again or something). They can't have anything as real and solid as a mother adopting a daughter. Because that can't be as real as a father adopting a son or something. Almost because it's too healthy, since Babs seems to genuinely take people like Cass in to help her rather than out of some need of her own. (Though I guess Dick maybe gets a little closer to that if you figure he's training Damian for Damian's sake, but of course there you've got the holy Bruce-DNA thing going on.) Obviously Babs (and Dick) get something out of helping others, but it does seem to really be helping others to be healthier, happier people.