You have a legitimate reason for requiring an updated Appendix A.
The irony is that the only triggering event under the regulation is the opening of a new account. There, the person authorized to sign is the person opening the account. If you ask for an Appendix A under other circumstances you are going to manufacture your own requirements for who signs the document.
Gone or not, the member is still a signatory on the account until you get a resolution from the LLC that says otherwise, but that's the LLC's issue, not yours. If John and I are the only members and I have "left the building" he's going to bring in a new resolution forthwith.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.