Andean or Puna Flamingo > Chilean Flamingo
This is an interesting case. Both rare flamingos experience massive natural breeding failures, where 100s of doomed eggs or chicks could be collected with no loss to the wild populations. It is because of unpredictable drying and freezing of Andean lakes. Zoos supported financially nature reserves in the Andes, where they breed, at least for some time. But no birds reached Western zoos. Currently, less than a dozen birds of each species live and irregularily breed in Berlin zoo. Slimbridge has a flock of Andean, but seems not to make much effort to breed them. Apparently they are not even sexed.
Madagascar Pond Heron > Javan Pond Heron
Greater Adjutant Stork > Marabou Stork
Madagascar Sacred Ibis > African Sacred Ibis
There are breeding groups of all three species in Western zoos, and it is difficult to explain by anything else than lack of interest.