Why not let anyone attend any public school? Because public schools are largely funded by local taxes. So a school in an expensive neighborhood with high taxes will naturally be a magnet for everyone else who can be a free-rider (in the economic theory sense). School vouchers fix this problem by carrying revenue along with the student instead of tying it to geography.
School vouchers with dollars attached to kids is effectively "choice, as long as you can pay for it". It's the status quo of zip code based schooling access, on steroids. That's exactly the "systemic issue" I'm referring to in my original post.
The claim that school choice would improve access to quality education for students in poor-performing schools is a complete farce.
I'm not saying that our current system works well or that it's particularly just. I'm just pointing out that vouchers make the tie between wealth and educational access even more explicit and codified than it already is. A voucher system would deepen, not alleviate, the inequities in our educational system.
Secondly it fixes incentive structures. Good schools in high-tax areas can now see inbound students from elsewhere as partially subsidizing the cost of the school, rather than being a pure cost center. Again this doesn't perfectly equalize anything but it's a step in the right direction.
Allowing low income students more freedom in how they spend their government-allocated funds mitigates the problems of wealth disparity. This is already how the university system works so assertions that it is a complete farce should also justify the abolition of the FAFSA system in exchange for federal funds that can only be used at the university nearest your home.