No. You can't take a voucher anywhere. It has to be accredited. Otherwise I could take it to an ice cream shop and call getting a cone an education. Just because you don't see the voucher doesn't mean economically things aren't equivalent, because they are. They just simplified the red tape of telling them where you are in a database, saying your voucher is available, and giving them authorization to redeem it.

That's what a "publically funded school is". They obviously have other funding like direct tuition payments, research grants, sports events. They are structured as an entity separate from the government. They manage their own funds that aren't the government's. They almost always have histories of foundation that were separate from the government. The government doesn't tell them who to hire or fire or who to appoint to which position. They basically are private institutions... that are eligible for receiving vouchers. Yes, you can take your voucher elsewhere. I can transfer to another state school and the new school will receive the funding associated with giving me an edumacation. An "actual" voucher system would yield the same result, just with a lot more interaction.

Also, the claim that it's not the same because it's only a partial coverage / reduction of tuition fails to consider that if you take a public school voucher to a private school you probably will still be paying some amount of tuition remaining. It is a private school after all. They aren't going to give a superior product to you with no price differential to using the less effective option. Vouchers would be partial coverage.