Nothing wrong with writing them down, but put them on paper and keep them in your wallet or a safe. We put our SSN cards, driver licenses and passports and credit cards in our wallets. Why not our passwords?

Isn't "Never carry your social security card in your wallet" standard advice? I was taught that since I was a child, before identity theft was even really a mainstream thing.

As to credit cards, if your wallet is lost or stolen, you can quickly cancel the cards. It is probably more difficult to quickly change all your passwords.

Keeping them in a safe, however, is probably a fine idea. Potentially a good one if you want people to have access to certain accounts if something happens to you.

because if you lose your phone, all your passwords are compromised.

The thief wouldn't know where the passwords go, nor my username.