We do have that freedom of choice. It just isn't paid for with taxpayer money. I can never understand this argument. No one is stopping you from offering your child an alternative or religious education. They're just asking that you pay for it. 
Common sense shows that we can barely afford public education for all as it is, we don't have the money to fund both public and private education. 
Part of the reason Alaska's education costs are so high is trying to run so many small schools spread out all over. Spreading out the $$ spent in all regions would only exacerbate this problem. 
Also, most of the people who support vouchers are also the ones complaining about the money spent on education right now. Vouchers would increase that amount by expecting the same amount of public buildings be open while sending their funding to private schools as well. 
As far as religious education, are you okay with your tax dollars paying for a Jewish or Muslim education for others?