The answer to birds getting out of the cage is to put signs around the sides of the cage. You don't need to get them to fly in, simply wait for them to land and plunk the cage down over them. Since they're not aggroed, you will have some time to get the signs around the sides before you start hitting it.
The cage also works brilliantly to capture horses, wolves, lions, and tigers. Other animals just pop out of the top.