An exhibit can be well-designed using multiple different approaches to layout, and as such, I'm not opposed to any way of organizing a zoo as long as it's well done. There are good complexes themed around reptiles, or primates, or big cats, and then there are also lousy ones and complexes that are relics of a different time. Likewise, there are both good and bad examples of South American, African, and Asian exhibits. Furthermore, claiming that this is a dichotomy where zoos must choose one of these layouts is false. While some zoos do choose to arrange exhibits purely in one method, others are very successful with utilizing multiple different ways to organize their zoo. Many of the major zoos, at least in the United States, have exhibits dedicated to both geographic areas, taxonomic designations, and other themes entirely.
Not mentioned yet in this thread, but something I personally am very fond of, is biome-themed exhibits. A lot of educational concepts that apply to a certain biome on one continent also apply to that same biome elsewhere. Rather than have strict Amazon Rainforest exhibits, a zoo can build an exhibit that is simply "The Rain Forest" or "Tropical Forest" and it allows a lot more flexibility species-wise, incorporating species from South and Central America, Africa, and Southeast Asia. Likewise, I find Desert-themed exhibits to be criminally underrated, and wish more zoos would incorporate a desert-themed exhibit into their zoo. Similarly, another great exhibit idea is behaviorally-based concepts. Not the most common any more, but nocturnal houses are a great example of this, featuring animals active at night and the adaptations associated with that. Other ideas in this regards would be concepts themed around arboreal behaviors/animals that climb, a "flight" themed exhibit pairing birds with bats and flying insects, or exhibits themed around a diet (for example, if a zoo built an exhibit themed around insectivores, including birds, mammals, reptiles, amphibians, and invertebrates that all eat insects). All of these are unique ideas that deserve consideration or inclusion by zoos when designing new concepts, and all have great educational opportunities within them.
Furthermore, and this is similar to
@JVM's point about species not fitting "obvious biogeographical spaces", I find there is a need for zoos to have either standalone exhibits or exhibits themed around broad concepts (ex. adaptations- all animals have adaptations so could be included). Some species tend to be not included in zoos, or are kept by very few zoos, because they can't neatly fit into a box for a standard theme. An example of this would be Japanese Macaques. They aren't super common in US Zoos, which I feel is a shame since they are fascinating primates and one of the few that are cold tolerant. However, they don't neatly fit into a biogeographic zone (Japan doesn't have many species present in US Zoos), and often aren't included in primate-themed regions due to their different climactic needs. I wish more zoos would incorporate this incredible species into their collection, but doing so would require some creative or different theming, such as a macaque-only themed exhibit (such as at Lincoln Park Zoo), an exhibit themed around a broader concept (such as Central Park Zoo's temperate themed region), an exhibit that is seemingly randomly placed/not in a themed area, or getting creative with new and original themes that can fit various other species, for instance incorporating macaques into an exhibit themed around the world's Islands.
A lot of smaller zoos also don't have themed areas, and I don't have any problem with this either. As long as the exhibits are well-designed and placed logically (not thematically, but physically placed logically, i.e. there's a logical path to take around the entire zoo), then I don't necessarily find that all exhibits need to be part of some themed realm, and there is a benefit to zoos having areas that exhibit everything but the kitchen sink. Theming can be really cool when done correctly, but isn't essential to designing a good layout!