> Moreover, password managers do not work if you use multiple devices for log in
I use a password manager with multiple devices, and it works. And yes, my passwords are "protected", that's the job of the password manager.
A classical password manager reads an encrypted database. In theory, you could upload your password database (usually just one file) anywhere, and wouldn't need to worry, assuming, that you chose a sufficiently long password for decryption, and assuming, that the encryption does not have weaknesses, which would allow an attacker to decrypt it without the password. In practice, of course you still wouldn't upload your password file to a public place, to reduce risks in the future. But anyway, the idea is, that only you know the master password for the encrypted database and so no one else can read your passwords.