The US Federal Constitution, back in 1787, immediately authorized a government-run postal service. If a similar scenario was echoed today, I think it would/should contain a government-run identity service.
Governments already have a compelling interest to identify people for the purposes of the legal system, property ownership, etc. With all that happening anyway, might as well have an API that allows for attestation and Single-Sign-On.
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P.S.: Not having it isn't really an option, since it's a void that will still get filled, just differently... Either with a hodgepodge of half-broken systems, or an abusive private monopoly, and no accountability or good appeals process.