There may be some "lore" carefully handed down from generation to generation in all of this. Broadening Al's suggestion, if your state has specific documentary requirements you must meet before you pay funds to surviving beneficiaries I suggest you find them in writing somewhere before you attempt to adhere to them. I've done state specific deposit administration programs in more than 10 states and not one of them had any statutory mandate as to what was acceptable documentation of death. What was acceptable was always determined by the institution that would actually be liable for distributing any funds in error. That makes sense.
Generally, it's like most things in banking, a risk based decision. In some community banks, no documentary proof of death is required. They take the fact that they saw the individual being loaded into the hearse as adequate proof. In others, the requirements range from a copy of the obituary, a copy of the death certificate, or a copy of the funeral directors' application for a death certificate. Obviously, some do require a certified copy of a death certificate, but that may be a bit of "overkill."

As noted, a certified copy may be expensive and may take weeks to obtain.