Frankly, I'd argue the opposite: zoos won't survive IF the collections become too homogenized. If this does happen, people would only ever visit their local zoo- meaning zoos would recieve less visits from tourists, less people visiting two zoos near their house (because if both are the same, why bother visiting both?), and just less to distinguish between them. If by phasing out species you mean ones that are almost gone already, such as asiatic black bear, mhorr gazelle, etc., I can absolutely agree with that. But if you mean phasing out species that are merely struggling, but still have hope, like andean and sloth bears, masai giraffe, goral and urial, then I strongly disagree with you.
I would definitely not suggest phasing out masai giraffe as we will need a giraffe population that is not made up of hybrids someday. I am not ignorant enough to support hybridization that isn't supported by a conservation organization. Goral and urial are probably among the mammals I am least informed on so would not comment there.
You bring up Andean, sloth and Asiatic black bears, and bears are exactly what is on my mind right now for a major example -- as aging bear exhibits in the country have needed updating and renovation, the number of species any zoo can hold has decreased. Denver has gone from four species to one, Lincoln Park from four to one, Milwaukee from five to one, Brookfield from four to three... Asiatic black bears are declining rapidly as the population is geriatic, polar bear habitats are being built to double as grizzly habitats when the population crashes, and the former sun and asiatic black bear spaces are expected to be used for Sloth and Andean bears going forward. A decision was made that not all of these species can be supported and clearly most institutions will be able to hold fewer bear species.
It would actually make the most sense for the AZA to focus on Sloth and Andean bears given that North American bears are not threatened at the level where ex-situ conservation is needed, but I do think the polar bear situation is hopeless.
And that's even without tackling the discussion why many species are struggling. For large animals, space and resources may indeed be limited. Elephants, gorillas and hippos are expensive. But for pheasants? Geckos? Rodents? Cichlids? Maybe the problem there has more to do with a lack of interest, creativity and innovation, and not with actual space. Many of the aforementioned small specialist collections are outperforming large zoos when it comes to those species. If a tiny zoo with a fraction of the budget of an average city zoo manages to make insects, reptiles or small mammals appealing enough to make make people visit only for them, why can't a large zoo find a way to integrate them into their collection?*
* Spoiler: I am convinced they can. Many of them just don't do it.
You seem very well-informed which I am not, forewarning -- but this is part of what has altered my perspective recently. When I was younger, it seemed like zoos were moving into a trend of making ecosystem-based exhibits which would include reptiles, amphibians, small mammals as well as larger species together based on a shared theme. Brookfield created many excellent exhibits based on this principle in the 90s/00s, with popular species such as alligators and otters anchoring the Swamp building which has reptiles, amphibians, even insects, or Habitat Africa including free-flight birds, mongoose, and kilpspringers indoors while giraffe and antelope roam outside. The popular species anchor creative display for other animals from their environment.
This style seems to have fallen slightly out of favor now - reptiles and so forth jump around terrariums in designated, taxonomic display again, small mammals feel increasingly absent or bottom priority, and multicomplexes become dedicated entirely to provide state of the art care for a single charismatic species while lacking secondary habitats. They act more as 'ambassadors' for absent animals than 'anchors' for smaller ones. There is also an understanding that sometimes spaces may need to be repurposed by animals not native to the exact range or habitat it was themed under. Both of these developments make the anchor species ecosystem concept feel dated at worst and trivialized at best.
I know it doesn't sound like it, but I am agreeing with you -- there is a lack of interest and (in terms of these smaller animals) creativity and innovation right now, in addition to the existing effort to whittle down larger species to improve husbandry, pointing to a continued decline in species held and total holders for species already competing.
That is why my perspective has shifted so much personally. The approach I might prefer no longer feels viable.