Bette's crush was never dangerous like that, but she's had that characterisation long before Obsession. She *wants* significant relevance to Nightwing, even if she doesn't have it.
I guess I can see a case for dropping it as she matures, but Flamebird is a solid name with a long history that doesn't get much use. The alternate she's used, Hawkfire, just sounds like bad word association from Flamebird.