A reputable crematorium would insist on a death certificate.
I’m not sure if having a death certificate filed triggers end of pensions though.
They are usually filed with Social Security.
Funeral homes will usually update social security. I don’t know if the county clerks (in my state) would update the database.
I know I had to inform the OPM and MA state teachers pensions for my mom.
Grab-backs suck when there is only a little money left and the person dies at the end of the month.