Unless you are talking about low-cost zoos like wildlife parks (and even they often have favorites like wolves, otters, bison, ...) you do need some sort of popular animals. Does that mean you always need elephants, giraffes, rhinos and lions? No, in smaller collections people will still come if those aren't present. In larger zoos that's a different case as people have expectations, you don't need every species but at least some of the most popular megafuana are needed. You clearly see in zoo reviews that many people do want to see them, and that they actively seek out zoos with these animals to visit from time to time. Some people will also visit zoos that have less/no megafauna, but this is rather an A and B situation instead of an A or B. Megafauna is and always will remain popular for the general public, so not to use this would be a waste.
Back to other collections, the most traditional megafauna isn't required but even they do need some popular animals. A good example I know is Nordhorn, a small park that has leopards, bison, wolves, harbor seals and zebras as more popular animals alongside a nice collection of lesser known species. A unique group are non-mammalian specialist parks. La ferme aux crocodiles seems to work, but they probably have additional sources of income besides the entry-fees. Bird parks like Avifauna and Walsrode do seem to struggle, and Avifauna is now housing some more popular animals like red panda, lemurs and South-American monkeys. So it's not impossible to not house any popular mammal, but it's very difficult to accomplish and probably only possible for a handful of zoos.
On the other hand, only ABC animals are also not the most popular. People want to be amazed and surprised. ABC animals can draw in visitors, but XYZ species are just as important in making those people's zoo visit feel extra special. One cannot live without the other.