“That way, people without kids could sell their vouchers to those who needed them.”
Those without kids would not get vouchers.
That’s not what Friedman suggested. He said everyone would get two vouchers. People with more than two kids would have to enter the “Voucher Market” and buy additional vouchers for their kids. That way, large families don’t get subsidized by people who don’t have kids in public schools (e.g., retired people). If you send your kids to private schools, you could sell your public school vouchers in the open market.