Safe deposit box custodians kept entry logs long before BSA/AML was even a gleam in someone's eye. They did it for their own protection in the case of an alleged "mysterious disappearance" of the renter's property. Eliminating the log would be a pretty avant garde move in my book; it would enhance the custodian's risk without providing any significant savings.
Until your post, it would not have occurred to me that a custodian would consider omitting the log or that one of the reasons for retaining it would be AML considerations. Now, if the proponent(s) of eliminating the log even mouth the words "customer privacy" I'm convinced they have wandered into an area of the bank where they have neither experience nor expertise. Give them a lollipop and send them back where they came from.
I'm interested to see how others respond.
P.S. Check NY record retention requirements for safe deposit access. If there are some (very likely) that might be your best argument against the uninitiated that you are required to document access. (Actually, what it probably means is that if you have the record you have to keep it, not that you have to have it.) What it does for sure is make you look incredibly sloppy for not keeping records at all.
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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.