Are you just making an argument for the sake of arguing?
I'm honestly not. It just doesn't make sense to me. That's like running RAID on your computer and saying that you don't have anything important to back up as opposed to using the additional storage space and simply backing up important stuff in a cloud or an external media device.
Now do I need random passwords thrown at me from an old PC? Not really.
Why do it then if you don't need it? Why not.
Nothing against doing it for no real reason, but if it's for simplicity purposes, then going fully automated is the obvious solution.
I don't use online banking because I HAVE to go to the bank 2 times a month already, Im not sure how much you frequent online banking (nor do I care) but I can't seem to think it would have any more use than a couple times a month to maybe check balances or move money. The same stuff I can do while at the bank that I am going to go to anyway.
I'm not sure how your bank is set up, but the line to discuss financial matters is different from the line to make normal transactions. So, instead of waiting in two completely differently lines, potentially doubling your time at the bank, you can complete your normal transactions any time of the day from your pc or cellphone. Not only does that minimize your time at the bank, but others as well.
Also, I would question the necessity of those two visits a month. I don't know your investment or am I well versed in investments, but given this automated, quick fast and in a hurry society, I can't imagine a bank providing a service that requires that many visits to the bank. Please correct me if I'm wrong, because I know not, but with the bank cellphone apps that allow users to deposit checks with their cell phone, it sounds a bit counter productive. However, I do understand from a "job security" point of view, but big money makers tend to follow money trends that benefit them, not their employees.