If I didn't have a car, I couldn't have a job anyway.
Your mother makes the point; why pay for something when you can get it for free. If there was a voucher system where she could send you wherever she wanted, would she still have have chosen the same way? Maybe, maybe not. But at least she would have had a choice. With the present government run monopoly on education she had essentially no choice.
We did have school choice. There was 2 public high schools in my city. I lived in the district of one, but they were both about equal distance from our house. Most of my friends were going to the other one, so I transfered.
If we did have vouchers, I know that my parents would've never sent me to a private school unless I wanted to. They let me and my brothers pick what school we wanted to go to for middle and high school.