Exhibits incorporating culture doesn't make them inherently poor exhibits, and it can be done in a solid, tasteful manner. As you point out, many zoos have gone "theme park" so to say with this cultural theming and intricate level of detail. However, I'm not so sure that I'd consider that a bad thing. The Mahajarah Jungle Trek at Disney's Animal Kingdom still remains one of my favorite exhibits, even though it is quite literally part of a theme park. What's most important, however, is that the theming is done in a manner that is respectful of local cultures, such as the case of Kingdoms of Asia at the Fresno Chaffee Zoo, where leaders in local indigenous communities helped with the exhibit design and construction in an effort to make a realistic cultural attraction.
Keep in mind that as conservation-minded organizations, zoos need to reach out to the native peoples as well, as conserving animals without acknowledging the culture and lifestyle of indigenous people is not an effective way to make long-term impacts on conservation. One great example of a zoo doing a great job with this is Woodland Park Zoo and it's TK-CP, conserving tree kangaroos while also building partnerships with the people of New Guinea. Like it or not, culture is an integral part of conservation and I don't see anything wrong with zoos reflecting this in their exhibit design, as long as it's down respectfully and does not compensate for poor animal welfare and/or minimal educational value. One of my biggest gripes with the Mahajarah Jungle Trek, despite overall thinking it's an incredible exhibit, is the lack of signage in the exhibit- as it's important, even when culture and theming are integral to an exhibit's design, that education and conservation are still embedded in these exhibits.
As for your points about taxonomic exhibits and collection planning, I don't think it's fair to say that geographic or cultural exhibits can't incorporate small mammals, or can't highlight the birds, reptiles, fishes, and invertebrates of a region either. While there are certainly many examples of exhibits that ignore these smaller animals in favor of charismatic megafauna, I feel this does more to reflect upon the zoo's decisions regarding collection planning rather than the exhibit themes themselves. To use an example I know you're familiar with that does a great job of incorporating these underrepresented types of animals into a zoogeographic exhibit is African Journey at the Lincoln Park Zoo, with a number of small mammals, birds, reptiles, fishes, and more from the continent of Africa. I would love to see more zoos take this sort of approach, and focus on the entire ecosystem of a given region, including these smaller life forms. Honestly, I'd love to see some zoos take this a step further and also focus on the plants and other forms of life that call these areas home too, creating an exhibit that can truly encompass a given region, rather than just "here's a bunch of African savanna animals exhibited near each other", or "this temple has a bunch of Asian animals in it".
An issue with the execution of zoogeographic areas, however, is I find oftentimes zoos focus too broadly on entire continents, rather than narrowing down the focus to allow a more comprehensive look at a region. No "Africa" exhibit will ever be able to replicate the entire continent, for instance, as there's simply too much to accomplish for the amount of space that most zoos can dedicate to one exhibit complex. Sure, a zoo might be able to do a good job representing "Africa" if they dedicate fifty plus acres to it, but most zoos don't have the space or other resources for that. I'd much rather see some zoos, instead of designing a convoluted Africa complex or a generic "Savanna"/"Forest", focus on much more specific regions or countries. For instance, I'd love to see an exhibit focused on the Namib Desert, and the animals, both large and small, that inhabit the desert. An exhibit focusing on the Horn of Africa, an underrepresented biodiversity hotspot, in zoos would also be greatly appreciated.
This applies to other continents as well, with how many generic "Neotropical" areas or "Amazon" exhibit, why aren't there more zoos investing in Central America and designing complexes featuring animals from a different Neotropical Forest? Even within Amazon complexes, it'd be great to see some more narrow focuses, such as an exhibit on the Atlantic Forest or an exhibit focusing on Bolivia. While these are just ideas that I thought of that may or may not be feasible, the point remains that there are plenty of regions of the world seldom highlighted in geographic and cultural displays, so it'd be great if zoos would look for underrepresented regions for their displays, rather than the same old, same old.
I've said this before and I'll say it again: There are two ways that zoos are able to distinguish themselves. The first is through their collection, and the second is through their exhibitry. As a greater focus is made towards sustainable populations, and as such the homogenization of certain collections that follows, it would be great to see zoos use exhibitry as more of a way to distinguish themselves, by focusing on themes and executions of complexes that have never been done before. Unique exhibitry can be just as exciting to see at a zoo than rare and unique species, so it'd be great if more zoos capitalized on this and planned more unique exhibits- whether they be geographic, taxonomic, or otherwise.