Those two cases you cite were about colleges and federal money and civil rights issues. As others pointed out, it fails to address tax money collected by a county authority which is given back to an individual to use at a school of choice.
Also, as I point out, the private school would simply not accept voucher students if any government thinks it can force changes in policies.
All states take fed-ed funds. In return, all states have agreed to fed-ed education goals. The states require local districts to conform with state law and regulation which is written to conform to federal law and regulation.
As a wise school administrator said to me on this topic: "If you take the king's shilling, you become the king's man."