I honestly don't know much about housing vouchers at all, but doing some google-fu for the last 15 minutes it seems like the problem is not so much discrimination against the PEOPLE that want to pay with vouchers, but the landlords are fearful of the IMPLICATIONS of dealing with voucher-holders.

In other words, landlords do not want A) to wait for the government to send an inspector which can take god-knows-how-long before the person can actually move in and rent, B) wait for the government to pay the landlords which, as with any aspect of dealing with government agencies, can take a dawn's age resulting in late payments or insufficient payments from the housing authority, C) endure yearly government inspections which, while designed to ensure that voucher holders do not live in substandard housing, are completely subjective and leases can be terminated because of a hairline crack in a light switch faceplate, &c.

In other words: it seems like landlords' problems aren't with voucher-holders, themselves, but the voucher system itself because it's administered by a government agency. In this case, the NYCHA which, everyone knows, is an absolute steaming pile of dog shit. It's an agency that would rather spend billions defending lawsuits than just making repairs in the first place.