I doubt it would've thrived anywhere as it was a specialist collection, no great apes, lions, tigers, giraffes etc etc etc... Banham has a lot of 'ABC' animals including giraffes, zebras and 4 species of large cat.
I echo Daves sentiments about missing out on something special. I was only young when it closed and didn't even know it existed, around the same time (as part of the family) I visited Banham and Suffolk Wildlife Park, which is now Africa Alive. Talk about regrets?
(In my defence I was about 6 or 7 at the time...

)
A few species in the 1990 guide book- Tayra , Grison , Paca , Tree-climbing ( grey ) Fox , Dusky titi , Black-bearded Saki , 7 different Spider Monkeys . 18 species of Marmoset/Tamarin - Weddells' Tamarin , Geoffroy's Tamarin , Tassel-eared Marmoset amongst others . 8 species of cat - black Geoffroy's Cat , Jaguarundi , Pampas Cats , Oncilla . Allen's Olingo , Naked-tailed Armadillo , Giant Anteater ( a rarity then ) . there were birds but mammals were the speciality .
I reckon the closest we have these days to Kilverstone in terms of collection range and focus would be Hamerton; one reason I am so fond of said collection despite its flaws. But even Hamerton has a handful of "ABC's", so this is one difference.
As for the species list you specifically quote, Bele, it is worth noting the captive state of the various unusual taxa listed:
Tayra - these have gradually been increasing in numbers as of late; something I am very happy indeed about.
Olingos - neither species is extant in Europe, with Kilverstone being the last holder of the Allen's Olingo in the 1980s, and Kerzers in Switzerland being the last holder of the Northern Olingo in approximately 2005.
Geoffroy's Cat - quietly chugging along in about 25 collections or so, although I think the population is in decline as I haven't heard of any successful births in recent years.
Jaguarundi - about 20 collections in Europe have these, and there have been a few births in recent years.
Oncilla - not doing amazingly well, with only 5 collections in Europe still holding these and no births in several years.
Pampas Cats - entirely absent from Europe since Parco Faunistico La Torbiera in Italy went out of them in 1996 (interestingly, they were also the last collection to hold Marbled Cats at pretty much the same time!)
Grison - the greater grison is down to one collection in Holland, the lesser grison is entirely absent from Europe as of 2000.
Paca - the Lowland Paca isn't doing too badly, but the Mountain paca is only just hanging on in 2 collections, one on the Isle of Wight and one in Russia.
Grey fox - entirely absent from Europe. Interestingly, ZTL has no record of Kilverstone having held this species.
Dusky titi - entirely absent from Europe since Kilverstone held them.
Tassel-eared Tamarin - entirely absent from Europe since Kilverstone held them.
Weddell's Tamarin - just about holding on at one collection in the Netherlands.
Black saki - just about hanging on, being found at Colchester and Belfast. However the latter collection recently managed a first European breeding, so maybe this species will begin to get a firmer foothold in coming years.
Naked Tailed-Armadillo - interestingly, ZTL has no record of Kilverstone having held this species, which has not been found in Europe since Jersey went out of them in the late 1960's.
Giant Anteater - having something of a renaissance of late, with almost 50 collections holding the species. Of the 12 UK collections holding the species, all but London have obtained them in the past 5 years or so.