It's a theme within the first three games, and to some extent Dragon Quarter, that the Wing Clan began to lose their clan abilities due to the Nina of Breath of Fire (1) marrying someone from outside the clan. In the centuries between one and two, that left the entire clan unable to turn into giant birds, and by number three only a handful of members of the clan possess wings... mostly the royal family...
That's about were my brain twitches from a perceived logic falsie. I've always had a different theory about the decline of power, and I'd like to finally lay it out for others to pick apart.
What stands out for me is that if the power decline was initiated by intermarriage of one member of the royal family, why is it that the royal family retains their power the longest? I don't remember all details of the game exactly, but I'm pretty certain it also had to be a member of the royal family that underwent the transformation into the giant bird form as well in BoF 2. If anything it appears to be a little foreign blood that is helping the royal family retain their power... which might make a lot of sense if the assumptions that Nina married Ryu in BoF (1).
In Breath of Fire 3, the goddess is revealed to have an agenda against power that doesn't stem from her. She presents this agenda with a motherly coddling front, but... well she really just appears to be a control freak. I could very easily see her leaching off the blessings of the various clans, with the expectation of the Dragon Clan which she had exterminated. Endless decline with age after all, and by the time of Breath of Fire 3 she is well past the millennia mark.
How she does this is as unanswerable as asking how magic really works, but a clear theory as to why she doesn't leach from the Dragon Clan as well presents itself; they still have their god. By Breath of Fire 3 he's nothing more than a voice in a mural, but he's still a tangible force.
As I said, this is an old theory of mine... like back from when I played Breath of Fire 2 for the first time old. But for me it's always made sense. The goddess holds a grudge, and at the end of Breath of Fire the armies of Windia swooped in to save her heroes from the collapsing fortress. That alone is a weak argument, but I've only found more pieces as the games went on...
...but in the end it's mostly my perception reinforcing itself. Feel free to tear it apart... or not.