I think you mean “If you want your API copyrightable, don't make it public”, not “If you want your API non copyrightable, don't make it public”

Now, how do you let third parties program against that non-public API? Would only showing it to licensees be a legal way to do that?

If so, and if the API becomes popular, I don’t see how to prevent those licensees from leaking the API to the world, say through small code snippets on Stack Overflow.