In order to change the password, you have to be logged in. So the website will have to redirect the user to a log-in form, passing along the change-password form URL when authentication succeeds. Then the user can put in the old password and new password, go through an optional MFA hokey-pokey, and get it changed.
If the intent is to speed up password changes, a few optional additions would be faster than the above. The spec could optionally allow (1) the account ID, (2) the user ID, (3) the old password, and (4) the new password. The response could be a challenge and consent request for the user, which the user could then affirm and submit.
The website could still dictate how this works, but the idea is that the password manager would pass along all relevant data in the initial request, eliminating the need for the user to enter it all manually, and eliminating extra page loads. But it requires no site-specific state on the client-side, because all requests would be exactly the same, to this generic URL. Whatever implements the spec URL on the server-side would perform the login and present the password reset challenge, pre-populated for the user.
This is not even an API. It is merely a proposal that wherever your change-password form lives, people should be able to get to it via this specific path, too. Perhaps this whole thing is a terrible idea, but either way the problem they're trying to solve is different than the problem you're trying to solve.