The short certificate is evidence that the estate was probated and not all estates need to be probated, so I wouldn't require it if the balances fall beneath $10,000 and the estate was not probated.
For close out of decedent accounts when the estate has not been probated, we require a death certificate, proof of the paid funeral bill, and an affidavit completed and signed by the claimant. State guidelines refer to payment order being spouse, then children, then parents, then siblings.
The affidavit states who is claiming the funds, their relationship, states that the estate was not probated and that the funeral bill was paid. I found the affidavit on a county website.
If you would like a copy of the affidavit, PM me your email address and I'll send it to you