21:53 here again. We actually are a Jewish family at St. Andrew's . OP, there is a big difference between independent schools with a little religion, and what you may be thinking of in terms of the parochial schools of our childhood with nuns. Students in St. Andrew's, St. Albans, NCS's, Sidwell (Quaker) etc. are all part of the same circle of independent school kids in the DC area secular schools of GDS, Bullis, Maret, etc. Yes, a kid needs to deal with a 20 minute or so Episcopal service each week, but these are usually accompanied by more universal themes, speakers, etc., and other religions occasionally (at St. Andrew's, for example, there is an annual chapel focused on the Jewish high holidays and a holocaust chapel as well). If you a very observant family, then no private will work for you except Charles E. Smith Jewish Day School -- there are many activities on Friday nights or Saturdays, including sports, homecoming, school musicals, etc. But if you are not that observant, I know some kids have developed a stronger sense of Jewish identity attending schools that are not secular.
I also think most of the publics have cut back on Latin and either don't offer it or only offer the first couple of years of classes. So that might take the magnets off your shopping list altogether. Good luck.