I wish it were as you paint it, Lazs.
The parent may then spend the money any way he wishes so long as it is at an accredited school.
Bang, you're dead. This is the crux of the matter. Accredited means you follow the state guidelines. The state is allowed to control education, like it or not. Remember, all powers not expressly given to the federal government belong to individual states. States are in competition with one another to produce the best schools (though it is unclear to me what standards are used to define "best"). States do not have to, nor will they relinquish control of the educational process. Blaming it on the teacher union (of which I am not a member BTW) is a red herring. The states follow the mold of the federal government of controlling all areas deemed public welfare- and education is certainly one of those areas.
States have fewer requirements on private education because they don't affect the standing of public schools. The state doesn't fund the private schools, therefore isn't accountable to the parents of the students in those schools. The second the state becomes accountable for private education through supporting them with tax dollars, you bet they will get concerned darned quick.
Vouchers are indeed controlled by the state. The state isn't going to hand them to just anyone for any reason. There will be strings, common sense tells you that. Don't think you're going to get an educational rebate and be told "spend wisely". There's the nasty problem of education being a requirement by law until a certain age, and parents that are irresponsible with checks can wind up costing taxpayers a huge amount of money (guess where they go to school after they blow that check? Someone will pay for that education). There is simply no way the states will ever hand hard cash over to the public, nor will they merely allow people to keep the money and pay for education. Chances are, the public would never even see the money, it would merely pass from state coffers to institutions of choice, and then only if the institution was satisfactory to state mandates.
Getting the picture?
It isn't a myth that vouchers are state money- they are. Right now a portion of your property taxes goes to public education. Vouchers merely take an amount that would have been spent on your child and allow you to spend that money on a private school. It is at best a tax rebate, except with strings. That money must be spent on education, and it must be spent on an institution that meets state standards. Well... there you go, the state calling the shots all around.