Thematically, Batman and Huntress are still inherently linked. He saw his parents get murdered in front of him when he was eight, she saw her parents get murdered in front of her when she was eight. Apart from that, DC's mostly playing the connection to show her as a shadow archetype, but I think it's still acknowledging that link. Sometimes canon also asks the question whether she's unlike Batman or (too much) like him, as well.
All that only works for the themes, but changing Bruce and Helena being related meant that DC was going to have to sacrifice some aspect of their relationship if Bruce having biological (adult) children was out of the question. Helena being younger than Batman and having been a vigilante for less time than he at least gives some legitimacy to Batman attempting to order her around.
I would argue that Huntress is part of the Batfamily if not the inner circle, even before Oracle got involved, simply because she was an active vigilante in Gotham and her relationship with the (rest of the) Bats went beyond the strictly professional. Calling her Huntress means that she is part of the family, imo.