The great part about poetry is that it is your perspective that matters most. If you want the bird to be indiffernt to its freedom, it can be. You would be using the poetic device of personification, which means that you ascribe human thoughts and emotions to non-human objects (or animals, in this case.) A really famous poem about a bird is "Ode to a Skylark" by Percy Bysse Shelley. In that particular poem, he praises the bird's song for its ability to inspire and soothe, but in the course of the poem, he gives the skylark all sorts of human thoughts and emotions. Another famous bird poem you could look to for inspiration is "Ode to a Nightingale" by John Keats.
I think what you have so far sounds really interesting, and it might give your poem even more depth to choose a specific bird to write about, like a sparrow, dove, crow. Different birds have different connotations, like an eagle makes me think of freedom, but a dove represents peace. You could give even more meaning to your poem through the type of bird you choose.