I think this one needs an out-of-universe answer. Yes, it totally makes sense if adult woman Barbara were a real person to want a mature name, and since there’s already a Batwoman, she’d need to come up with something else, maybe not even bat-related. In-story, remember, the “Batgirl” is a playful costume she whipped up for a masquerade, and blurts it out in-the-moment to protect her identity - so she never really sat and pondered. But out-of-story, she was created to appeal to (younger) female viewers of a TV show, which was already very successful with “Bat-branding” everything, so it makes sense to stick with that. (Would “Batlady” or “Batqueen” or even “Batwing” (“The Batman Nobody Knows” wouldn’t come along until 1973) have worked just as well? Probably. But they went with Batgirl. It’s certainly catchier, and has precedence, if you leave out all the maybe-sexism-but-maybe-not logic…) Once she’s named, as @CaptainYesterday mentioned, the marketing money is pumped in to the name, and it’s very hard to change it. (Many characters have changed names, but once you’ve got public recognition, it’s tougher.) Reboots give writers a chance to retcon in a better explanation for why an adult woman is calling herself a girl.
This is true even without the TV link – new characters who are introduced under a protégé or in tribute to another character with name recognition have consistently sold better than those who are introduced fresh. Less people are going to pick up “The Librarian”, even if it’s exactly the same story and art that’s published under “Batgirl”, which is likely going to be sold on the shelf right next to the “Batman” title they came in for.