If you have access to a copy, take a look at Brady on Bank Checks, ¶¶ 13.01 - 13.07. You'll find a long list of reasons a bank must be careful when handling checks payable to itself (along with some cases in which failure to do so came back to bite the bank in the assets).
Incidentally, one other situation where a check payable to the bank would be appropriate -- deposit of federal tax payments with IRS-supplied coupons. Many businesses now use electronics and ACH to make these deposits, but a lot more still use the old coupon method.
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John S. Burnett
BankersOnline.com
Fighting for Compliance since 1976
Bankers' Threads User #8