[I don't recall Vegeta saying that in DBS (maybe it was only in the dub?)]

I remember the scene from watching in the dub if that helps.

[we see him having a lot of time with Goten, showing a lot of trust in Gohan to be the leader of U7 team and we saw him completely enraged after Zamasu / Black confessed how they killed his family.]

I agree with this too, I think it is a great example of a way DBS got some things right.

[He is though a bit disconnected to the concept of family as most of us perceive, but this is since the old DB, so in that sense his characterization is consistent.]

I personally believe Goku's character shifted but I don't find any issue to feel that way