Everything you do in granting access to a box is intended to reduce the potential for an unauthorized access.
If your employees actually follow your existing procedures, requiring them to write down what they saw is truly pointless. It also creates a scenario where, if you require them to note ID, but then you have an occasion where an employee fails to do so, that becomes evidence against you where none would have existed otherwise.
There's a point in safe deposit operations where well considered caution becomes paranoia...
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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.