When a school presents itself as Catholic, but the reality of the school's culture undercuts the Catholic faith, that does more harm to a kid's faith than being an openly secular atmosphere.
Example: My husband taught at a Jesuit high school The athletic director was an ex-Catholic who regularly testified to kids how he'd not met Jesus until he left the Church. And of course he was in no risk of losing his job because the school had a fabulously successful athletic program.
The head of the Theology department had a degree in chemistry and knew nothing to speak of about theology (I share your confusion. In a Jesuit school? But aren't they.......Yeah.)
One of the priest faculty members had a last name, that, with the change of one letter could be transformed into a slang word for a male sexual organ, and he gladly and jokingly referred to himself this way to students.
And you want to pay (at least) $6,000 a year for this?
In my own school, 2/3 of the faculty were not Catholic and most of them, while not hostile to Catholicism, couldn't and weren't interested in really understanding and sharing in the Catholic mission of the place. The principal, while effective in some respects, fully planned to reduce theology classes to two or three days a week before he transferred to head the Episcopal school. The new principal promptly appointed an ex-Catholic Mormon as Dean of Students.
Granted, not any one of these things destroy the Catholic mission of a school, and there can be other teachers, administrators and programs that compensate...but when you have decisions like that being made, they're usually only the tip of the iceberg.