If you write a seed phrase or words we can't consider it as brain wallet but a paper wallet because we can only call it brain wallet by memorizing the seed phrase. That is why Brain wallet is a bad idea. 12 to 24 words is not easy to memorize so having a hardware wallet is the best option without memorizing the seedphrase.

I think you just don't understand or confused about the concept of brain wallet.

Better read this https://en.bitcoin.it/wiki/Brainwallet