With pheasants (and also some other intensively-managed gamebirds in aviaries such as painted and brown quail) the problem stems from removing the eggs for artificial incubation generation after generation. It seems to be an accumulative effect. It doesn't help that pheasants are stupid. The same thing can be seen in some cichlids in aquariums, angelfish being especially renowned for it, where they simply eat their own eggs.
It will be interesting to see how Auckland's flamingoes deal with raising chicks in the future when none of them were parent-reared themselves.