The average user is not savvy enough to sign consistently, but can keep a private key secure? That's a big assumption.
For those in the US who don't want a national ID: it wouldn't have to be one if the states issued their own. Yes, this would mean they'd all need the technical capability to operate a CA securely. Fund them enough to do that.
The US desperately, desperately needs a proper solution where "thing you give tons of people" isn't also "exactly enough to steal your ID".
As a curious factoid, in-between the world wars, it was popular for workers to have their ID number tatooed on their arm.
Using cryptography in addition to "classic" means of attestation sounds a lot better to me. You could also make it an "optional feature" of people's citizenships (like 2FA on an account) to ease adoption.
The user just needs to learns how to perform some simple operations with the app, they never have to know about private/public keys.
I'm sure some service would figure out a way.