Hello.
I've got to be honest: I don't believe most recently extinct animals would be that remarkable
The thylacine itself became notorious because it went extinct, in the first place. Depending on how easy it would be to breed them outside Australia and New Guinea, they could have become one of the more common Oceanian species in zoos, mostly due to their sheer size.
The quagga was cool-looking, of course, but I'd say the black-and-white pattern other zebras have is more striking to the general public than Equus quagga quagga's. I'd say their presence in zoos would be dictated by the ease of keeping them in mixed-species exhibits. I'm quite sure if they were still alive, Karoo-themed displays would be way more common.
Bubal hartebeest would likely have the same fate as other subspecies of Alcelaphus buselaphus, and be rare (if not absent) from captivity. Bluebuck could also be part of Karoo exhibits alongside quaggas. Maybe they would become as common as roan and sable, but this is debatable.
The extinct populations of tigers in the Sunda islands of Bali and Java were way smaller (and therefore not as remarkable) than their mainland counterparts, so I'm not sure if they'd make it to capitivty in large numbers. I've read somewhere at least Bali tigers have never been kept in zoos in the first place.
Had not most of the notable fauna native to the Indian Ocean islands next to Africa (Rodrigues giant tortoise, dodo, Rodrigues solitaire, Réunion night heron, broad-billed parrot, Mauritius gray parrot, Seychelles parakeet) vanished, exhibits centered around them could become a thing. If not, they'd likely become extensions to Madagascar exhibits.
Labrador ducks would likely end up becoming just another species of waterfowl raised by privated breeders. On the other hand, like their living relatives, the extinct grebes would be rarely found in zoos.
Great auks wouldn't be as popular as penguins, I'm quite sure. As with other seabirds, they would not be common either.
I'm not a specialist, so please don't take this too literally.