You can restore all bitcoin address you have only with 12 word mnemonic, so it's not necessary to backup all the private keys. You better make sure to backup/remember those 12 words instead.
But, backup the private keys if you still feel anxious

Hello, can you give an example of a situation when BTC owner MUST have private keys? In other words, a situation when 12-word recovery phrase won't be sufficient?
For forks. Not all seed-protected wallets support exporting private keys, e.g. Ledger Nano S.