I don't think having to sign into other websites is that much of a bother, nor that people are that motivated to talk to their bank or physician on a regular basis that would drive adoption of this sort of thing.
And in return you have to stick all your eggs in one basket, get what would probably end up being a single persistent online identity that goes under your real name (if it's tied to an email address you use for business stuff), and that's owned by a company and may not even be willing to give them back to you (would you even own the private keys if it was being implemented on the server?)
