Allowing any student to enroll at any public school regardless of their home zip code would be a much more meaningful form of choice than vouchers.
Why not just let anyone in a metro area attend high school at the richest/best high school in the area?
If you can answer that question, you can pretty well predict what systemic issues would be caused by a voucher system.
I'm generally skeptical of mixed public/private education systems with partial choice and subsidies for private options. We have exactly such a system for higher education and it's a fucking disaster.