I'm sure you will disagree with me, but, as intelligent and educated as you are, I'm also sure that you can understand where my reservations come from, if not the ultimate conclusions that I draw. My district has worked to revamp the assignment process. Each family is now given a group of schools within an attendance zone that they can choose from. Each school chooses a theme. If there are more applications than the school can accept there is a lottery to see who gets in.
Administrators then try to give the parents their second or third choice.
If everything is full, they are assigned to the school that has spaces. You might be outraged by that, but there is a very real limit of spaces in private, voucher, and charter schools, and not everyone would be able to get in, even under your plan. You'd sent the leftover kids to a public school whose funding has been gutted. I hope you see what I mean about the fairness issue. Hope your eyeballs aren't bleeding from reading all this.