Having had the privilege of working with a close friend of the Leopard Man of Peckham, I feel moved to suggest that any purported public purpose of zoos is, in fact, secondary to their private purpose: delighting zookeepers. That is to say: people who love wild animals and want to work with them find justifications for the public maintenance of zoos where they can work. (Or, alternatively, find resources to set up a private zoo, which Todd Dalton did after his menagerie outgrew his back yard.)
Indeed, I would go further. All or nearly all cultural institutions of value are producer- rather than consumer-driven. Someone does something because he or she is mad about it, and is sufficiently good at it and/or sufficiently persuasive about it and/or has sufficient natural charisma that he or she pulls the public along as consumers/spectators/participants. We have paintings because some people love painting, and we have museums because some people love collecting. I don’t think zoos are any different, really.
To paraphrase good old Friedrich, “Whatever is done for love always occurs beyond public purpose.”