Not all birds have to be 100% caged, especially if they're the only animal in the house. Just be careful opening and shutting the door.
With birds, though, you may need to get two. Birds can get very lonely. They always sell lovebirds in pairs, but a good pet store will explain which ones need a buddy and which ones are okay alone. (Cockatiels need a friend. African Gray parrots are intelligent enough to bond with a human instead, much like a dog.)