I think they made Kate and Bruce cousins to make Batwoman more marketable - as judging from her original intro in the 52, and then Elegy through to the first solo, I get no sense of them being related or even knowing each other that well. The Kane name was left-over from the previous version of Batwoman. I know that they gave Martha the Kane name, but that is also a left-over of the old version of Batwoman who was Bruce's aunt. Also, the original Batwoman was gone for a long time, so they probably gave Martha the Kane name for the same legacy reasons they gave it to the original Batwoman, not planning at that stage for a new Batwoman to be created.
Then, when there was a swap of writers for Batwoman, and the Zero Year event happened, suddenly not only is Batwoman appearing in the crossover events for the first time, but Kate and Bruce are first cousins. Not much more is said about it, until the Tynion 'Tec run, and then it's hammered in again and again, with not only other members of the Bat-Family commenting in order to remind audience (most notably Tim and Babs), but multiple flashbacks of Bruce and Kate as kids together. Fast forward to the end of last year, and with the Batwoman TV series on it's way, and Kate's TV story matches in with that of Supergirl's of getting-out from under the shadow of their more famous older cousins. With Kate as heir-apparent to the cowl, being family is a quick and simple explanation, and presumably a faster way to get straight to the bat-action with all equipment provided.
I haven't seen any explanation for why the Kane Manor is said to be in ruins (seemingly for years) in the Tynion 'Tec run, because in the 52 when Renee goes to visit Kate it's at the Kane Manor, from what I recall, and they even had servants - so the place was not in ruins.
Also, yeah, Bette has invisible parents. She is also apparently invisible to Bruce, since apart from the one Zero Year issue, they seem unaware of each other's existence. Although, until the second Batwoman solo, Bruce and Beth didn't seem to know the other existed either.
The Family dynamic creates some problems, but since they have leaned so hard into Kate and Bruce being related, unlikely there's any going back anytime soon.