A fundamental American ideal is that we are all created equal and therefore  deserve an equal opportunity to succeed based on our individual merits, rather than the family, race, religion we were born into. 
I would support taxpayer funded vouchers for private schools but only on one condition: that students in every public school receive equal opportunity, i.e. per pupil funding equal to that in the most expensive private school that accepts vouchers.  So if a private school has per pupil funding through donations, tuition, and vouchers of say $25k, every public school in the state would receive $25k per pupil.
Of course, "pro-choice" folks don't want that.  They want even greater inequality of opportunity by taking money away from public schools to fund voucher programs for wealthier parents.