It is possible to build private zoos, but they will not be up to the level of AZA city or state funded zoos, and even the ones that are close (or previously even AZA) tend to avoid it these days as AZA has strict rules on financials and private sales. There are a number of decent private zoos that will not be AZA but work with AZA zoos but also sustain their operations on gate revenue and animal sales (Iguanaland, Brights, Sylvan Heights). Some of these are well regarded, some mixed.
A few of the best are specialist facilities rather than typical zoos, Sylvan Heights, Iguanaland and Reptilandia come to mind. There are some general zoos such as Wildlife World, Metro Richmond, Gulf Breeze, Brights that are decent to good in most of their exhibitory. I know WW is controversial here, but they are a really unique private zoo given their massive size of land and incredible collection, not to mention the inclusion of a full size aquarium; they have been steadily replacing the worst cages over time.
Then there are private zoos with decidedly mixed facilities such as Bee City. Alabama Gulf Coast Zoo opened a completely new zoo in 2020 that is quite good in a new location. The newest of these that I am aware of is Broadway Zoological Park in NC,
@Astrotom3000 went there earlier this year.
All that said, safari parks seem to be the "go to" for new private generalist facilities with more than several having opened over the last 10 years. Texas Zoofari, Alabama Zoofari (the same company), Atlanta Safari Park, Georgia Safari Conservation Park, Wild Florida (they had the walk through before, but the safari park area is pretty new), Aloha Safari Park, etc. Some of these have walk through parts that would qualify as a small traditional zoo.