Who Should Be Permitted to Provide Public Schooling?

Hess says that traditionally, public schools are deemed to be those directly accountable to elected officials or funded by tax dollars. Hess also points out that it's getting harder and harder to define what a public school is! I agree. With the advent of vouchers and with different levels of government involvement, some supposedly private schools seem to be more "public" than conventional public schools.
I'm not so much concerned with labels of public and private as I am with just making sure that every American family has a fair opportunity to put their children in safe schools that offer a quality education. Some of our hard-earned tax dollars go towards education. If a parent applies for a voucher to get what they feel will be a better education for their child, I don't have a problem with that. I do believe that we need to make sure that vouchers are used responsibly, and that the voucher system is not abused.
I believe that there are good and bad schools on both sides. In one of the articles I read, the author states that a school is as good as its administration, staff, admission policies, class size, parent-support, and budget. I agree. Much like an athletic team/organization--quality starts from the top down.
I do have an issue with "fairness". Like a golf or athletic country club, private schools are the domain of those who can afford them. The biggest difference between public and private has always been income level. Vouchers may make private schools available for the poorer families, but what about middle class families who want a different choice? I also feel that if any school is receiving state tax dollars, than they should have the same rules, regulations, and level of accountability as the public schools. Some private schools want public dollars, but want to still run the school by their private rules. If we (meaning: charter, private, public, homeschool, etc.) are all competing for students and families, than we should have a more level playing field. Finally, to receive public money, the private schools should have to broaden their rules for admission. Public schools cannot turn away anyone, regardless of handicaps, behavior, family income, etc.
In short, I believe there is a place for all kinds of schools (labels), but if we are indeed competing for students and funding--than we all need to be on a more even level of accountability, rules, and regulations.