Correct, but most people just don't care that much about deer. When they've seen one species, they've had enough deer for one zoo visit. Sad but true.
Want to make the eyes light up of the average zoo visitor? Show him/her a reindeer/caribou and remember to say something about Santa Claus.
Secondly, most South American deer have never been widespread in European or North American zoos. A number of Pampas, marsh,
Hippocamelus and brocket were kept many years ago (mostly 50+ years ago) and more recently keeping of these have typically been restricted to very few (often only 1-2) zoos per continent. Such tiny populations almost invariably end up disappearing due to inbreeding or other issues. Today all that remain are brocket: A few brown brocket in Madrid but it will also disappear unless new blood is imported. Old thread about the red's in North America:
Red Brocket Deer
So, for a European/North American zoo to get more they'd need to get into contact with institutions in South/Central America/Mexico (esp. Pampas and various brocket, mostly the widespread species, are kept in some numbers in their native countries), do the work and pay the price to import, and unless it involved a fair number they'd end up with a small population that eventually will disappear again. A lot of work and money for a type of animal that most zoo visitors don't care much about seeing.
White-tailed deer is primarily a North American species, but its range also extends into South America. The North American (not South American) subspecies are widely kept in zoos/parks/ranches there, but relatively few are kept in Europe. It would be easy to source more from North America for a European zoo, but this leads back to the issues I mentioned above.