Sorry for replying with an outside-USA answer, but I remember seeing the three species at Berlin Tierpark.
Correct, my friend. They did use to have all three (Chapman's, Grevy's and Hartmann's Mountain Zebras) although I do not know whether this is still the case or not.
I'm not your friend. I know they used to have all three species. Like you, I don't know whether they still do.
Sadly Marwell no longer has plains (Chapman's) zebra just Grevy's and Hartmann's mountain zebra. Apart from Marwell and London (long ago) the only zoos where I recall seeing three zebra species are Berlin (Zoo and Tierpark) and Paris (Vincennes).
I'd imagine that Hartmann's Mountain Zebras would be the least common in the USA, so the best idea would probably be to find which zoos have that species and then just check if they have either of the other two as well. For example, Disney's Animal Kingdom has Hartmann's Mountain Zebras as I found with a quick Google. Then I found an article (from 2012, so not necessarily still valid) which said they had acquired a herd of Plains Zebras, and amongst that herd they also had one Grevy's Zebra, Zebras return to Disney's Animal Kingdom Members of the herd are 5 to 10 years old, weigh between 650 and 800 pounds, and are all females. Disney has no plans to breed the Plains group, Hohne said. However, it has one female Grevy's zebra, an endangered species, that has been living with the Plains zebras. It's possible the company eventually would try breeding Grevy's zebras, in cooperation with the AZA, Hohne said. There also are zebras — the Hartmann's Mountain variety — living at Disney's Animal Kingdom Lodge.
As far as I know, DAK is the only place in North America that keeps all three species. If another zoo does, it is a very recent development. Also @Chlidonias is right to imagine that HM zebras are the least numerous here; their population is less than 60, compared to ~175 Grevy’s and 250+ plains. That’s also just AZA and associated private facilities; there is also a large number of plains zebras in private hands around the US.
I would argue that Maneless is the least attractive. Anybody remember 'Animal Life' magazine, that packed up in the mid 1960s? They had a long-running advertisement for Hartmann Mountain Zebras, £2000 each, supplied from South Africa. I always wondered how many they sold.
I always enjoy seeing the maneless zebra at Colchester because I see that form so rarely but I agree with you that they are the least attractive. Yes, I used to read "Animal Life" magazine as a child back in the 1960s although I always preferred the rival publication "Animals".