While it is true that some elite high schools do not offer AP courses because they feel that their courses are better than AP courses, that may not necessarily be true for most high schools that do not offer AP courses (or alternatives like IB etc.).
Given that this private high school is new, it is hard to tell which is the case here. Classical languages and literatures do not have AP courses, so the absence of AP courses in those areas is not surprising. But does the school have students are who advanced enough in other subjects like math and English that they would want to take AP courses in those subjects?