Is it safe to store seed phrases (private keys) in a bank vault? If written on a piece of paper, would the recovery seed phrase or questions for accessing a collection of web wallets and hardware wallets be safe to keep in a mini vault drawer at the bank? Do bank employees have access to vault drawers somehow, i.e. in the event of natural disaster, or are there any procedures for opening vault drawers that might compromise the words on the piece of paper?
Given the above, also how safe is it to additionally store a copy of the same paper at another branch location as a back-up?
The best thing to do is use a secure passphrase in addition to your 12/24 word phrase, and keep the passphrase in a separate location, or even no location at all but your head and maybe share it with one or more family members.
Using BIP-39, the same 24-words with a different passphrase will create completely disjoint wallets which cannot be linked by outside observation. You can keep reusing the same words and have different wallets by using different passphrases. Make sure passphrases are a reasonable length and complexity and not dictionary words or birthdays. However, brute-forcing these is difficult because PBKDF2 is used for hashing, so you don't need to go overboard and use a passphrase you might forget.
It might be a good idea to include a small amount of bitcoin in the wallet using your 24 words, but with no passphrase. This can serve as a 'decoy' wallet, which can be monitored for activity to discover if your 24-words have been compromised. It also serves to allow plausible deniability to the amount of bitcoin you own, because if you are under duress from a person (the State for example) trying to force the information out of you, they cannot possibly know how many wallets are created from the same 24 words. You can plausibly claim that what is in the decoy wallet is all the bitcoin you own.
Since both the 24-word phrase and your passphrase are necessary to access your funds, the security of the 24-word phrase is not as paramount as without the passphrase. You should keep a copy on a steel plate to resist natural disasters, and keep a written paper copy in another location which is more accessible to you, but which can be safely disposed of with fire.