But the places that want to know your identity for casual surveillance will not settle for just a pseudonymous FIDO exchange. What will happen is there will be a handful of companies that you directly authenticate to with FIDO (eg Google), who require a bunch of personal information to create an account, that then vouch for your larger identity to third parties.
The problem is not technical - there are many ways to replace website auth passwords that would be better. The problem is the business desires all but guarantee newly developed methods will leak more of our personal information. Is a business more likely to do the work of implementing FIDO themselves, or to tie into Sign in with Bigcorp? Given what is already happening, that answer is clear.
Simple per-site passwords continue to exist due to the momentum of everyone understanding them, despite their drawbacks for the surveillance industry. Regardless of what other methods are offered, we should aim to preserve the existence of the password method as at least one of the choices, such that we can freely decide whether those other methods actually benefit us. The growing insistence on snake oil "2FA" is already bad enough.