I think the issue here is that once you get to a certain level of respectability and accountability as a public collection, you can't be seen to be purchasing rare animals. OR importing them, which is still happening, so these zoos are restricted to working with other zoos, or bringing in new species in conjunction with in situ projects (eg Owston's civet). The vast array of small mammal species that have been quite successfully bred as house pets by private breeders are not as accessible to collections not so ready to be uncovered acquiring animals for hard cash. It does still happen, but not nearly as much as it used to.
I am 80% sure that Amazon World was set up by a major bird importer from the mainland. Look at the range of species they have. There is a reason for that, and it's not because the major zoos of the world are all rushing to send them rare animals on breeding loan. They have more toucan species than all the other UK zoos combined...it's not because they're great toucan breeders. As far as I know, all their birds are still kept indoors with no access to outside flights.
Tamanduas are bred, hand-raised and kept like house pets all over the US, as well as Paca, Binturong, Sulawesi macauques, Spider monkeys, and many more species you'd expect to only really see in zoos. Google 'pet tamanduas' and be prepared to cringe at these animals wearing anoraks and clambering over armchairs in people's houses. Yet they seem to do well and breed in domestic situations. Fennec foxes, similarly, do amazingly if you treat them like domestic dogs, but breeding in zoos is still pretty rare. Private breeders need to be brought into the picture with EAZA and regional studbooks/programmes. I don't know why it isn't worked into the wild animal licensing system. It could be a mutually-beneficial arrangement, but I fear many zoos would be horrified at the 'humanising' of animals they send to breeders, as would many breeders protest the hands-off approach of many zoos.