yeah but what the Big Daddies were, their function, their creation, even their day-to-day lives - we saw all of these first hand in the first 2 Bioshocks.
The only thing we know about Songbird is its motives: Keep Elizabeth in tower/Get Elizabeth back into tower.
There's even audio diaries insinuating that there might be a human inside Songbird, Big Daddy-style. And, if that's the case, knowing what we know of Big Daddies, and knowing that Songbird is "programmed" to feel "incredible betrayal" whenever Elizabeth isn't around, I'm betting that Songbird's biological components are *another* Booker, kidnapped from a universe we never saw, and Songbird's instincts are a forcibly warped version of Booker's own paternal love for Anna!Elizabeth.
Which is also why we probably never got much about Songbird in the game proper. There'd really be no way to impart this without having to prematurely pull the ripcords on a bunch of other reveals.
Hopefully it'll be DLC.