But through rounds of rejiggering and three phases of funding — each with its own maze of rules and requirements — Mr. Heller’s group built a team to distribute aid, whittled down the processes that delayed it and ultimately concluded that the best way to help was the most straightforward: Give the money directly to renters. In response, more than 400 state and local governments have used money from the federal CARES Act to set up funds to cover at least $4.3 billion in rental assistance — money that has helped tenants pay their bills and landlords stay current on their mortgages, according to a database set up by the National Low Income Housing Coalition, a policy group. Each layer was ostensibly created to improve programs — preventing fraud, making sure the money went to the neediest tenants — but added numerous hurdles for both tenants and landlords, and in the end cost time. 