I don't love that shift — which definitely occurred during O'Neil's editorial reign, especially in Knightfall/Prodigal and beyond — and I think it mostly hurts the characters. I feel like the New 52 was onto something with framing "Robin" like a prestigious internship, more than as a bunch of adopted children. It's a mentorship.
Actually I think the Marvel Cinematic Universe version of Spider-Man/Iron Man might be the ideal version of the Robin/Batman dynamic, for me... I don't think those movies capture "my" Spidey, who I prefer as more of the Lee-era rebel/loner than someone who would want to be anyone's protege... they're better Robin movies than Spidey movies, imo.
Batman/Iron Man is paternalistic, and is a fatherly figure, but doesn't need to be housing, clothing, and feeding. Still "Family" but they're not literally dad/son, adoptive or otherwise. Because if Batman is more literally Robin's dad, then basically he needs to be able to say "I love you" to his children. But of course he can't because he needs to look cool, so instead it becomes like this weird Fonzie story point where because Batman is so cool he's slightly broken.
That's why I like the 90s Tim Drake Robin so much, because he does have that extra bit of independence with Jack being his Aunt May. Batman doesn't "own" him. It makes it a bit more of Robin choosing to go into this on his own, rather than inheriting the family business.