Be aware that if you have ever sent any bitcoin transactions from your Bitcoin Core wallet, then the wallet created hidden addresses that it didn't tell you about, and stored some of your bitcoins in those new hidden addresses.
If you want access to all of your bitcoins in the future, you will need ALL of your private keys (including the keys for the hidden addresses).
Every time you send a new transaction, the wallet will create another hidden address with another hidden private key. Therefore, if you write down ALL your private keys, and then send a transaction after that, you will no longer have ALL the private keys.
You mean you need the private keys to all addresses that contain bitcoins. Otherwise they are lost.
But shouldn't a backup of the private keys create private keys for the change addresses too? If not then this sounds like a security risk.