While I am not one myself (though I don't eat that much meat), I'd also like to point out that a fairly large percentage of keepers and other zoo employees are vegetarian or vegan, probably a greater proportion than the general public.
A few thoughts on folks who are opposed to zoos, the motivations of which are highly varied, and should not be seen as a monolith:
1.) Some are just lazy, want to feel like they're helping animals, but don't want to actually expend any time/effort/money. Regardless of what you feel about SeaWorld and the welfare of captive orcas, for example, it's an indisputable fact that SeaWorld has no negative impact on wild orca numbers. They aren't taking animals from the wild, and even when they were, it wasn't in any appreciable numbers. Still, people can tell themselves that by NOT going to SeaWorld (and let's be honest, most of the people who jumped on the Blackfish bandwagon weren't going there anyway), they're making a positive difference to help whales. Substitute zoos/elephants and you get the same feeling. Sure, you may buy a shopping-cart full of palm oil products, then drive home to your house in a gas-guzzling car, and make several other decisions which negatively impact animals in the wild, but you're helping animals by not going to the zoo.
2.) Some of the most stringent animal rights folks I've met have been folks who have been exposed to the worst-case scenarios of animal abuse and neglect, such as working in high volume shelters, where abused, confiscated, and mistreated animals far outnumber the happy, healthy ones. It's easy to take a dim view of human ownership of animals in such a setting, and can easily lead people, subjected to so much misery and death, to prefer a world without captive animals to one in which so many are treated so poorly
3.) There are fair philosophical discussions that can be had between the benefits of zoo conservation and a large scale and the well-being of individual animals. A person who prefers the personal liberty and autonomy of an individual animal to the overall well-being of a captive population has different priorities - priorities that I disagree with in some cases - but they aren't inherently wrong or misguided for thinking and feeling the way that they do.
4.) Zoo folks are not a monopoly. I've worked with plenty of keepers who work with some species, but are opposed to the keeping of others. I've known plenty of keepers who would happily see a world with no elephants or cetaceans in zoos (which we don't mention too often, because elephant keepers can be an intense, rowdy bunch... on zoo director I know who played devil's advocate with a group of them at a conference barely escaped with his life...). And going on that, it's normal for zookeepers to have doubts themselves. I know I have. I've worked in some sorry little zoos with some bad management, but even at the AZA zoos I've worked at, there have been times I've had doubts about whether we're really doing an adequate job. There's been more than one animal that I've worked with to whom the last words I've ever said have been, "I'm sorry, you deserved better from us." (Ironically, this is why I feel like animal rights folks make bad animal keepers - if you convince yourself that all captivity is inherently bad and a cage is always a cage, it's very hard to strive to do better, because you're convinced on some level that it's all equally horrid. I'm just as exasperated by the keepers I know who are convinced that they know everything and can do no wrong).
As a final thought (and one I've posted elsewhere on ZC), a very small percentage of the public are actually strongly anti-zoo (though we tend to pay the most attention to them). An equally small number are obsessive zoo fans. The vast majority of people are fair-weather friends, with us in the good times, against us in the bad, than swaying back. Look at the enormous vacillation in public sentiment aimed at Cincinnati between Harambe's death (May 2016) and Fiona's birth (January 2017). The trick seems to be staying on the good side.