the ban is on birds in general, not flamingos specifically. There was an application to get flamingos added to the import list about ten (?) years ago but it didn't go through apparently. The ban on importation of all birds and avian material (e.g. feathers) was introduced in 1949, excluding imports from New Zealand. The ban was extended to include New Zealand in 1971. The main aim of the ban was to try and prevent the introduction to Australia of exotic avian diseases. There was a short-lived reprieve, when bird imports were allowed between 1990 and 1995 but then suspended indefinitely due to renewed disease concerns. Currently the only birds allowed to be imported into Australia are live domestic pigeons; the eggs of domestic chickens, domestic turkeys and domestic ducks; and live king and gentoo penguins from New Zealand (as of 2008, due to them coming from a contained facility in New Zealand to contained facilities in Australia).
Actually, what Australia needs is more zoos willing to display more birds, non-native or otherwise. I think Adelaide may be leading the way here. Hix