I think this would need a total re-think before the masses could use.
I'd love to see browsers implement it by forcing people to store client certs on a USB-key or a phone by default. I think some kind of physical item that contains your keychain would be much more intuitive to many. Everyone is familiar with mechanical keys, they know not to leave them around, they know that they need them to unlock things and they know if they lose them they need to replace them.
The analogy to mechanical keys doesn't hold up under closer scrutiny, though, right? House keys don't serve as an identity, just an access mechanism. You don't lose all future access to your house if you lose your keys; a locksmith can independently verify (with reasonable certainty) that it's okay for you to obtain a new set of keys which give access to the same house.
I feel like the human predisposition to risk aversion is even more of a factor preventing adoption among average users than poor UX (not to mention lack of awareness). What do I tell my parents when they ask "What if my computer crashes? Would I not be able to log in to the website and see my stuff? What good is using a website if I can't access it from any computer?"
Until something as securely portable and loss-resistant as one's own memory is achieved, I don't see passwords being less popular than any other access mechanism for the average user, no matter how significant the other downsides.
I'd love to see browsers implement it by forcing people to store client certs on a USB-key or a phone by default. I think some kind of physical item that contains your keychain would be much more intuitive to many. Everyone is familiar with mechanical keys, they know not to leave them around, they know that they need them to unlock things and they know if they lose them they need to replace them.