It's really hard to do that and make the thing a consumer-friendly product. We've been trying to solve this problem for most of the history of computers, yet, attacking authentication (often indirectly) is still the #1 way that computers are compromised.
Most people simply are unable to properly handle private keys. All of the systems with the highest levels of consumer satisfaction have third parties that manage (or at least can override) keys on the user's behalf. Systems that do what you're suggesting are notoriously plagued with issues surrounding key management to the point where they never reach mainstream use. i.e. PGP, bitcoin, etc.
Most people simply are unable to properly handle private keys. All of the systems with the highest levels of consumer satisfaction have third parties that manage (or at least can override) keys on the user's behalf. Systems that do what you're suggesting are notoriously plagued with issues surrounding key management to the point where they never reach mainstream use. i.e. PGP, bitcoin, etc.