You're missing that there's no need whatsoever for "Flamebird," in this context, to mean anything other than "What the former Supergirl is now calling herself." The whole forgotten-god-of-Krypton nonsense would be a problem to tack onto Linda, sure, obviously. There isn't, however, any reason whatsoever to have invented it in the first place. "Flamebird" is known to be the name of an old Kryptonian hero. Linda's powers are appropriate to someone called "Flamebird," and Linda hangs with kryptonians. Her old name is no longer available to her.
Absolutely nothing would make more sense than her picking up that name when she comes out of retirement.