How safe is BitcoinCore since it is using owner's passphrase? When I want to create a wallet with BitcoinCore by going File-&gt;Create wallet, it asks me to choose a passphrase for it. Over the mouths exploring Bitcoin I learned that it is very unsafe to use custom passwords since they can be easily cracked (if I'm not mistaken, this type of wallet is called Brain wallets). I'm assuming that the passphrase will be used as a seed for a random generator in order to create private key for that wallet, and all this seems pretty unsafe to me.  Also I noticed that there is a checkbox saying Encrypt wallet. Does this mean if I don't make this check, my wallet will be unencrypted and therefore unsafe?
No, not all passphrases are used to generate the private keys for a wallet. Only specifically brain wallets do that.
Passphrases in actual wallet software like Bitcoin Core and Electrum are used to encrypt the wallet. The private keys are still randomly and securely generated. Your passphrase is only used as an encryption key so that the private keys are stored encrypted on disk.
It is safe and recommended that you encrypt your wallet with a passphrase. If you choose to not encrypt your wallet, then the private keys will be stored unencrypted on disk and any person or malware with access to your computer can steal your private keys and thus your Bitcoin.