> it’s also a big security improvement for all the cases where the third party need that info anyway Security there (in the cases I am guessing) is determined

> it’s also a big security improvement for all the cases where the third party need that info anyway Security there (in the cases I am guessing) is determined not by Alice being able to prove that she is Alice, but by Trudy not having secondary ways allowing her to claim that she is Alice. So, it would be «a big security improvement» only when login were restricted to needing a certificate.