I don't think it works that way. The family wouldn't be in debt for years or at all.

"Generally, the deceased person's estate is responsible for paying any unpaid debts. When a person dies, their assets pass to their estate. If there is no money or property left, then the debt generally will not be paid. Generally, no one else is required to pay the debts of someone who died."

Source: https://www.consumerfinance.gov/ask-cfpb/does-a-persons-debt-go-away-when-they-die-en-1463/