> a secret word for phrase that both you and the service you want to sign into know
That would not be a password, that would be something you share. An actual password must be possible to verify without it being stored on any device. On the service side, it's the same as for certificates, as you describe. The service can't store the password because that would invalidate it's usefulness as a way to prove someone is who they say they are. This is why we store a cryptographically secure hash code instead. It is also why the password hash code must be generated on the user's end, not on the service side. You never want to "transmit" passwords in plain text because transmission across the internet is an act of making copies of the data transmitted in the memory and storage of all the devices it transmits across. The moment you send a password across the internet, it is compromised.
So as the service, you don't know the password, you only know that the hash code you received matches the one for that user, and you are reasonably certain that there is no known way for someone to generate that hash code without knowing the real password. Therefore the person trying to login in must be who they say they are.
Passwords and private keys only work as authentication if no one else knows it, has possession of it, or can get access to it. If there is a flaw in any one of those aspects, then the system doesn't actually prove a person is who they claim to be. It only proves that a person is someone who knows, possesses, or has access to that thing. That might still count as evidence that they are authentic, but more will still be needed to actually prove they are authentic.