It really does stop. What can you do with someone’s SSN? Get loans, open bank accounts, receive government benefits, set up utilities, etc. It harms someone because creditors falsely believe that the SSN’s holder owes the debt, or the government believes that the SSN’s holder received benefits, etc.

People who are falsely reported as dead have a difficult time doing anything… certainly a hard time getting loans. It’s certainly going to be hard to make a claim against an estate that’s been closed for a couple years, with a debt that is dated after that person’s death.