Tinamous and rheas of course. I don't know how "loyal" pelicans, geese and swans would be, but those could certainly be options.
I know Wildlands' squirrel monkeys, for example, can fully leave the zoo grounds but don't because they want to stay close to their food source, and Avifauna does have a few fully flighted pelicans free-roaming, but they presumably stay in the zoo because of the rest of the (non-flighted) pelicans that have to stay in the lake. Beekse Bergen, on the other hand, has had issues with a few of their pelicans escaping very often.
Apenheul has free-flying macaws that technically stay close, but "close" in this scenario does include the entire city of Apeldoorn