The New Saigon initiative would be deemed by courts to have sufficient nexus with State power such that they act as an extension of state themselves. Such institutions and charter aren't uncommon in US. Most school districts are governed usually by kind of law you outlined. Under settled law, public schools act as arm of government, thus have same restriction on them as any other governmental entity has.
And before you come up with more contrived example, one has to keep in mind that laws are enforced by judges, who can usually see past complicated structure to determine what would be extension of government and what would not be.
