Can We Have Both School Choice and Shared Institutions?

I'm a decades-long supporter of school choice in nearly all its forms and likely to remain that way so long as traditional, district-operated public schools ill-serve so many kids, produce such widespread mediocrity by way of achievement, give parents so little say in so many matters, and cater to the interests and priorities of the adults in the system more than the needs and interests of children, taxpayers, and the general public.