My Mother died at age 86 1/2 on January 28th, 2020. Her February check was direct deposited on Feb 1, 2020 and then automatically withdrawn from her account on Feb 5th, by the SSA.
I thought they were very efficient. Not sure if they got the information from the Hospice service or the funeral home. But they sure withdrew it promptly.
Funeral home automatically notified Soc. Security Admin. in the case of my mother and father. I called myself just to make sure but they already knew.
I believe that the county/town clerk that registers the death updates Social Security.