A reduced budget does not increase costs.
It does decrease funding without decreasing costs, though.
On your second point, that might be true if the school voucher were for the total cost/student. But it's not. It's a tiny fraction of that.
Put another way, they don't actually help the poorest students attend private schools.
But besides that, again, so what?
So, I don't want my tax dollars going to private and religious institutions who do not answer to the taxpayers.
Keep your boss's boss off your boss's back.