The reason Babs does it here is that in this retcon she's reduced to a young girl looking for validation from one male authority figure after another.
Yes. I can't get past that.
Personally, I really hope nobody else pledges an oath. Why can't it just be Dick's special thing? It makes sense, and was kind of intrinsic to the story instead of being a thing to do just to do.
(Personally I pretend Batgirl Year One never happened.)