I think you may have your cause and effect wrong. Here is an example from my own life. In my county we have neighborhood schools and county-wide schools. Your kids will go to the neighborhood school UNLESS you apply to send them to one of the county-wide schools. Applying is not nearly as convenient so only parents that really valued education bothered to send their kids there. There was no entrance exam but the county-wide schools had measurably better academic performance. The kids were the same but the parents were not. Religious schools are the same, they are a self-selected group so it is to be expected they will perform well.My wife is a retired Lutheran school teacher. The Lutherans have had a "Protestant" school system in the US for over 150 years. The school where my wife taught and we sent our children always performed substantially better than the public schools in the district.
It cost a significant amount to send a child to that school. Mostly solid Middle Class or the more well to do could afford the tuition although church members got a break.
Without a voucher system poor families who wanted a good education for their children could not afford to send their child to the excellent school. They were stuck with low performing public schools.
Why do you Liberals hate the poor so much?