> Wouldn't it be more efficient to put the risk on the bank, so that it is the bank that seeks insurance to cover that risk?
No, because the value stored in the boxes differs by orders of magnitude and the bank doesn't even know the value.
$200/yr is just not a reasonable amount of payment to take on a 10s of millions of dollars liability, and most box customers wouldn't pay whatever price would make that liability reasonable.
(Based on the 500x annual cap mentioned in the article you could imagine that the bank would want $20k/yr for a $10m liability)
As a consumer, I have no idea what the risks of a safe deposit box are, and so have no idea what I need in the way of insurance to cover those risks.
The bank, on the other hand, has real data about the safety of their safe deposit boxes, and the value of loses when those boxes are compromised.