There are many private schools worse than public schools in nice neighborhoods. Most “Christian” schools and southern segregation academies fall into that category. There are some excellent Catholic schools, particularly Jesuit school, but many ordinary Catholic schools are mediocre. I have also tutored students who were very unhappy with schools that appeared good, such as a girls school in classy suburban area and an Episcopal school in a high end part of a city. It seems like some of these schools are geared towards rich kids who aren’t that good academically. IMO it is sort of a waste of money to go to a private school that doesn’t have really high admissions standards and intense academics.