In BftC, though, it makes sense for the Dick reaction Daniel set up that Nightwing wouldn't want to talk; he's supposed to be closed off and distant and stuff
See, I take issue with Daniel making him in denial and closed off in the first place. Because in Nightwing (with Tomasi writing) and in Robin (with Nicieza writing), it was Tim who was emotionally closed off and in denial, and Dick was the one who wore his heart on his sleeve and faced up to Batman's death and what he learned from Bruce during his life. Tim was the one who was distancing himself from his emotions and relationships, leaning toward becoming cold, obsessive and calculating.
I think as a character Dick is much more in touch with his emotions than Tim, so what we saw in those series made sense to an extent. Then Daniel starts off BtfC with their positions inexplicably switched.
saving him was her Establishing Character Moment, why wouldn't they have a connection?
Definitely agree with you there, yeah. And there's also the fact that it's now established that he knew Babs was Batgirl, so that's another thing which would endear her to him. Then there's the possibility that seeing Jim and Babs together may have been her first exposure to a healthy father-daughter relationship, and Cass's closeness to Babs also kind of makes Cass and Jim close by association. (Alfred is a kind of grandfatherly figure for Cass, so I can see Jim being one too.)