There is no “right way” to set up a housing voucher.
That is why Milton Friedman and other conservative economists proposed basic income instead of housing vouchers and hundreds of other benefits for the “poor”.
Any system of vouchers requires a huge bureaucracy to administer and monitor the system and is subject to an amazing array of abuses by both landlords and tenants.
“Universal basic income” is terrible, except that it’s much better than the alternative of specialized vouchers/entitlements/etc.
Housing vouchers will do nothing more than increase prices. Landlords know low-income renters will have that voucher, so raise otherwise market prices accordingly.