It looks like our current approach isn't working. What if we had each site publish its login/registration endpoints in a URL, e.g. .well-known/loginurls? Then the password manager could detect you're trying to register or log in and log you in itself, generating your password in the process. Why aren't logins machine-accessible yet?
Keepass does that pretty sure all the others also do.
But again what problem are you trying to solve? using password managers is easy as pie today including automating signup and generating passwords, most people do not use them.