Indeed that is possible, the problem is all the greedy corporations that want to suck all our data, so they can't simply trust the signed messages of Metamask and want instead to implement layers of poor software for their profit
oh i mean, identity/authentication alone doesn't make profit right?
The problem with using PK cryptography is that if you lose your keys, you'd be locked out.
Then we would probably have other key schemes M-of-N etc or social proof to compensate for the problems with single-key PK, which then has other problems.
The chrome extension to cryptography would help - but we have FIDO that does exactly that (and webauthn i guess).