That's actually not true. Vouchers are legally viewed as a source of income, which cannot be discriminated against in Washington DC. Now, do some landlords still do that? Sadly yes. And they can be taken to court by the applicant for some good money and/or accommodation if the applicant is wise to the law.
Landlords and property managers are required, by law, to accept vouchers.
That being said, DCHA (now split into two organizations) was so messed up that the application process would get hung up for months, during which time another person with money ready to go can walk in and apply to take the property. The landlord can continue to list the property until payment is received by an applicant, and it's easy to see arguments on both sides for discriminatory behavior.
It's all a mess.

Thats not true, DC is a income source blind city. If you reject due to voucher you can get sued.

A fair point, and if the overpayment were the only issue with this whole thing, then we could probably learn to live with that. Unfortunately, the problems run so much deeper than just that. Right now, DC has created a system that incentivizes landlords to cash in without providing any other essential services whatsoever, setting the entire living arrangement up for failure. Much like the federal student loan system where colleges don't care about your ability to afford college or whether or not the quality of the education is worth the price of tuition. They can jack up tuition as high as they want and phone it in knowing they get their check from the government up front. Both systems rely on overpayment with the good intention of expanding access to those who could otherwise not afford it. But without actually accountability, this is how they inevitably end up.