As for the applicant's current landlord, you have to take what they tell you with a grain of salt because they could just be trying to pass a bad tenant off to the next guy. 

I even had to pass a background/credit check just to get on the waiting list for one.  

My advisor (CPA and experienced landlord) included a clause in the Holding Fee Agreement that it would be refunded if the tenants failed to pass a screening or credit check.

I actually had to pass a background and credit check just to get on a waiting list for one of these places, and pay the application fee.  

Adjusting the payment standards happens on an at least yearly basis, and if market rents go down or the government no longer chooses to fund the program/chooses not to pass a budget, then yes, obviously rents can go down, but the first risk is universal no matter who you rent to, and the others are seriously problematic even if you don't have anyone on a voucher, provided funding is cut, reduced, not passed, for something related to your tenants.

@Steve Rozenberg right now the tenant pool is so strong that new tenants need to show they can pay the rent (job or plenty of cash), excellent credit score, pass a tenant screening w/o problems and be able to pay all the deposits etc. 

And it looks like it's completely legal to simply offer month-to-month agreements across the board.Next, they'll pass a law saying you have to have one-year contracts for all of your tenants. 

Sure, I'll be able to pass a fortune on to them when I die, but they will resent me for not being there when they needed me the most.  

I am sending them a rental application but they are likely not going to meet the criteria for credit or pass a background check.I live in Minnesota, what are my options?  

The postings are also very clear about our qualifications:- Income > 3x rent- Credit score of 600+- Must pass a background check and have no evictions in the last 7 yearsWe've had multiple open houses and despite probably 50+ inquiries online, have had almost nobody show up to the open houses.