Perhaps, wing-clipping may be the single most important factor in breeding failure. I would also suggest that enclosures not being netted goes with the former ... (no netting, no possibility for no wing-clipping).
I read somewhere that up to three quarters of male flamingos are unable to copulate after wing clipping.
The solution are netted aviaries (some Czech zoos build inexpensive and attractive walk-through aviaries), or leaving males free-flying as in Basel zoo, or controversial clipping both wings symmetrically.
I think, however, that zoos have much to do besides that. Too small flocks and uneven sex ratio (generally unsexed birds) seem to be the rule not exception. Also, many falmingo pools are in shade, too small, too close to visitors or otherwise unsuitable.
I would suggest replacing flamingos in small pools by eg. Black or Black-necked Swans, building suitable aviaries and pooling birds from several zoos into large flocks of 50+ for breeding.
I think wing clipping is also reason why birds like Marabou, Shoebill and Saddle-billed Storks don't reproduce in zoos. In contrast, swans, geese and ducks seem to live and breed perfectly when wing-clipped. Perhaps because waterfowl naturally become flightless during their moult.
cranes and pelicans too.
Andean flamingos at Slimbridge
I wonder how Slimbrige would feel about sending a few females to Berlin. On the other hand, aviary of Andean and James Flamingos in Berlin is very small, and may not accomodate additional birds.
About breeding in small flocks - the manual I linked has statistics of breeding in different flock size. All large flocks breed, but only few of small flocks ever try to breed. I never heard of regular, sustained breeding in such a situation.
I have been thinking on similar lines of thought as conservation dependency should be prime in seletion of those taxa deserving captive space. Besides their prolonged non-breeding status puts an onus on Slimbridge to actually sent off some of their birds off somewhere else where breeding conditions are met more fully.
I also do believe that Slimbridge might wish to go into aviary for these flamingos. Do they practice wing-clipping (which I think is also a major factor in non-breeding).