We agree that your bank should obtain some form of documentation that grants an individual the authority to act as an authorized signer on a trust account.
The “agent of trustee” form that you describe can serve as appropriate documentation for this purpose. We also think that you may use the same documents and procedures that you use for authorized signers for your other accounts; we are not aware of any special rules that apply to authorized signers for trust accounts.
Having said that, we like your bank’s current approach. Then if the trustee dies, the agency relationship automatically ends and the “agent” will be removed from the account, leaving it up to the successor trustee to reappoint that agent or another individual.