I had been thinking about this scene since finishing the game and I came up with a theory.
Warning: Spoilers Ahead
When Elizabeth first discovers the Lutece twins and comes across the two brooches, she is offered one of two brooches and leaves the decision to Booker. I came to the conclusion that by the end of the game, choosing whichever brooch didn't change the outcome of the ending in any way. Upon further analysis I realised that it wasn't the outcome of the choice they were presenting to us, it was the act of making the choice at all: It's just another of the Luteces' thought experiments. If you consider Elizabeth as a puppet, she can see all the strings, yet never be able to free herself from the inevitable. With Booker however, he is the one who makes all the choices (albeit minor) in the game's events before it leads up to the ending yet he doesn't know every possible pathway. This lead me to believe that even when Elizabeth sees the two brooches, she doesn't make the choice, you as the player make this choice and she has to go with your decision rather than make it herself. So it's not whether Elizabeth could be the bird, it's that she is, was and always will be the cage and you will always be the bird guiding the way, unaware of the consequences of your decision.
So in the end it begs the question to the player:
Would you rather be free and unaware of all the possible consequences or...?
Would you rather see and know every possible consequence past,present and future, yet not be able to make a single choice for yourself?
Needless to say, it still ends the same way.