Re the comments bout "discipline & dedication" being the reason private schools dominate.
I'll say this: If East Jefferson had the community support of, say, Katy, Texas, it would take a good chunk of the good players from Curtis and Rummel almost overnight.
Again, the secret in Texas is not that the UIL keeps private schools out, but that the UIL members aggressively fund and support athletics (and, in many areas, academics) in a way most private schools can't compete with.
Look at the stadiums they build. Whenever outsiders get outraged about the expense of those things, usually those districts brag about their academic programs and test scores, noting that they did not rob Pater to pay Paul with athletics.
If Jefferson Parish consolidated some schools, used the savings to add an aggressively ambitious IB and AP curriculum for the top students, vocational education (maybe more classes that can count as credits to Delgado or UNO for things like nursing) for average students and, importantly for this discussion, better facilities and more coaches for a more intensive athletic experience, then I bet they'd get a lot of private school students back.
But Louisiana school systems won't do that. Even in places like Jefferson and Ascension, where demographics aren't necessarily the reason for keeping schools "neighborhood," the preference is for many schools that offer the same, crappy, sub-standard curriculum instead of larger schools that offer a more diverse curriculum and things like bigger coaching staffs.