I don't think such unfriendly UX would ever be considered in such a mainstream service

You already have to login with a username and password to access Facebook, unlike whatsapp where your phone is the sole key. Facebook could just construct the key at logon time based on the user's password.

the OPTION to start an encrypted chat, with a chat specific password?

Or storing the key in localstorage?

To enable end-to-end encryption, you need more than local storage, you need local computation using local execution of signed and reproducible code. Javascript in browsers is fundamentally not such a platform, mobile applications are - to the degree you trust Google, Apple etc.

i guess by "each time" i was being somewhat ambiguous. it could mean each login, or each install, or each time the session clears.