I believe this is not only restricted to Catholic schools though they are the most common. Most religious schools have higher scoring students.
IMHO, we always hear about such and such school (system) has X% kids proficient with $Y/year per pupil. But what I would really want to know about a school is how does a year change at the school change the proficiency of the class. If the class of 3rd graders starts the year at 20% proficient at 2nd grade level, and ends at 22% proficient at 3rd grade level, that might be a good school, even though a single point in time check says 22% proficient. But the numbers we get aren't really useful for that; a cohort analysis would be better; there's real privacy implications, but that doesn't make the numbers we get useful. :P
I wonder if USA schools are similar. It's next to impossible to require belief.
You need to test to an academic standard of course, as they definitely want to keep the bar rather high. So they won't take all comers. But if you are either just starting out or come with an academic track record/high percentile test scores you shouldn't have much of a problem at all. When I went even 30 years ago there were plenty of low income kids who were not academic superstars. The only real metric that was universal across the board was the requirement for involved parents.
I'm sure other areas are different, but Catholic schools in my region have really suffered in recent years with a lot of them closing down.