1) A good public education should be available to all students, with reasonable transportation provided. This is an axiom for me.
2) Private schools do not seem consistently provide a better education, but they do enable siphoning and segregation of students. Students with disabilities of various types can be excluded; the education can be explicitly religious; private education can really enable exclusivity of many kinds. On one level all that is fine -- but I don't want taxpayer dollars to support it. Raise money thru your church to send kids to that stuff.
In addition, I think your premise is false. In the school district I live in, children can apply to attend any school in the district. Charter schools are available as well as public schools. Choice is available with a publicly funded model, and as a parent I am partaking in that and switching a kid from a high-scoring school that I'm nevertheless not fully satisfied with to another high-performing school that rivals the privates for about $30k less a year.
But in the end, I actually don't think competition forces improvement. The misalignment of incentives is too great. Individual teachers want to do their best for their kids, in general, and they also want to improve their income. To do their best, they try to help the visible high achievers if they can but they absolutely must take time on the kids acting out. Administrators have to deal with district edicts and teacher demands, but they don't have the money to support teacher development (which would be the #1 thing to help both teachers and kids). School districts/boards need to show they're "using technology" and "being equitable" and "being innovative" and all sorts of stuff that is not the same as improving outcomes. Everyone is supposed to do well on the tests, even when the tests are stupid (and whoo-hoo are some of them terrible). Schools have to deal with homelessness and hunger, because no other regional agency deals with all the kids in the area. Nothing lines up and so we end up with free lunch that's not so nutritious but does hit a calorie bar, attempts at equity without supports (putting kids with behavioral problems into crowded and understaffed classrooms), a bunch of teachers who are scared of math yet have to teach it, who would benefit most from upgrading their skills....