I imagine what you are proposing then is to record the entropy on the password when you first register and for accounts with sufficient password entropy to not ask for a captcha after few failed attempts.
With that, the site gives away whether the account has a low entropy password or not.
> I imagine what you are proposing then is to record the entropy on the password
Or just generate secure high-entropy passwords and force users to use them.
Making users look up SMS codes before each login is acceptable. Making them solve obnoxious, long, privacy-hostile riddles is acceptable. But forcing them to use pre-generated secure passwords?! That can't possibly work. They will revolt!
With that, the site gives away whether the account has a low entropy password or not.