> why would any of those people care if you have declared bankruptcy?

Landlords aren't in the business of providing housing for free, and seek restitution for damages and unpaid rent through the courts. A low credit score or bankruptcy shows that a potential tenant could incur significant damages and unpaid rent, and leave the landlord high-and-dry when the bill comes due.

In today's owner's market, there are plenty of other potential tenants without a poor credit history or bankruptcies.