The problem needs to be solved at a more fundamental level - people should not have to be forced to perform a function that they are demonstrably bad at. Mitigation strategies like having randomised passwords and storing them in a shared authentication platform are only masking the reality that passwords are a bad way of performing authentication.
Not that I'm clever enough to come up with an alternative mind you, and not to suggest that I don't agree with your premise that using a password in multiple places is a bad idea.
If you have given an untrusted third party site the credentials that you use on other sites, that meat is complete fetid. It is now deadly.
This whole discussion is arguing about what to do once the meat is rotten, rather than daring to maybe discuss not selling rotten meat in the first place.
When a site gets compromised and the passwords may get stolen (because of weak or no cryptography), the site should send out password reset emails en mass, and that should be the end of the whole issue. Instead it's moralizing about how they put everyone at risk because of other sites where the same credentials work. No, the user put themselves 100% at risk. But it is never discussed that way, and instead we continue this ignorance train.
As an aside, I marvel that some defensive imbecile keeps coming deep into this thread to downvote me.
Comments like this are the "rotting meat" of Hacker News. Please just leave them out of your posts.
The whole discussion revolves around a fundamental principal that is simply broken to begin with, akin to "How to try not to die when you eat rotting meat".
Don't eat rotting meat. Use a fridge. Etc.
In the case of passwords-
-Use a shared authentication platform -or on sign-up implore that your users do not use a shared password. Education -or offer, or force, a generated password
But instead we'll discuss the risk that shared passwords get lost, when they were in the wild the moment you used them on a second site.